\ 


'    iliii  iiiliir"''  iiiiiii   1 

P^EEXIPNS 

■  i     i 

i         I  I  11  ill 


1 


i 


IN  THE  SAME  SERIES. 


V  ON   THE   STUDY   AND    DIFFICULTIES   OF   MATHEMATICS.     By  Au- 

gustus De  Morgan.  Entirely  new  edition,  with  portrait  of  the  author, 
index,  and  annotations,  bibliographies  of  modern  works  on  algebra,  the 
philosophy  of  mathematics,  pan-geometry,  etc.  Pp.,  28S.  Cloth,  91.25  (5s  ). 

V  LECTURES  ON  ELEMENTARY  MATHEMATICS.     By  Joseph  Louis  La- 

grange. Translated  from  the  French  by  Thotiias  J.  McComtack.  With 
photogravure  portrait  of  Lagrange,  notes,  biography,  marginal  analyses, 
etc.  Only  separate  edition  in  French  or  English.  Pages,  172.  Cloth, 
$1.00  (5s.). 

^HISTORY  OF  ELEMENTARY  MATHEMATICS.  By  Dr.  Karl  Fink,  late 
Professor  in  Tiibingen.  Translated  from  the  German  by  Prof.  Wooster 
Woodruff  Beman  and  Prof.  Daznd  Eugene  Smith.     (In  preparation.) 


THE  OPEN  COURT  PUBLISHING  CO. 

324  DEARBORN  ST.,  CHICAGO. 


/ 


MATHEMATICAL    ESSAYS 


RECREATIONS 


BY 


HERMANN  SCHUBERT 

PROFESSOR   OF    MATHEMATICS    IN    THE    JOHANNEUM,    HAMBURG,    GERMANY 


FROM  THE  GERMAN  BY 

THOMAS  J.   McCORAJACK 


CHICAGO 

THE  OPEN  COURT  PUBLISHING  COMPANY 

LONDON  :  Kegan  Paul,  Trench,  Truebner  &  Co. 


Copyright  by 

The  Open  Court  Publishing  Co. 

1899 


^^^f^ 


TRANSLATOR'S  NOTE. 


'T^HE  mathematical  essays  and  recreations  in  this  volume  are  by  one  of  the  most 
■*■  successful  teachers  and  text-book  writers  of  Germany.  The  monistic  construc- 
tion of  arithmetic,  the  systematic  and  organic  development  of  all  its  consequences 
from  a  few  thoroughly  established  principles,  is  quite  foreign  to  the  general  run  of 
American  and  English  elementary  text-books,  and  the  first  three  essays  of  Professor 
Schubert  will,  therefore,  from  a  logical  and  esthetic  side,  be  full  of  suggestions  for 
elementary  mathematical  teachers  and  students,  as  well  as  for  non-mathematical 
readers.  For  the  actual  detailed  development  of  the  system  of  arithmetic  here 
sketched,  we  may  refer  the  reader  to  Professor  Schubert's  volume  Arithmetik 
U7id  Algebra,  recently  published  in  the  Goschen-Sammlung  (Goschen,  Leipsic), — 
an  extraordinarily  cheap  series  containing  many  other  unique  and  valuable  text- 
books in  mathematics  and  the  sciences. 

The  remaining  essays  on  "Magic  Squares,"  "The  Fourth  Dimension,"  and 
"The  History  of  the  Squaring  of  the  Circle,"  will  be  found  to  be  the  most  com- 
plete generally  accessible  accounts  in  English,  and  to  have,  one  and  all,  a  distinct 
educational  and  ethical  lesson. 

In  all  these  essays,  which  are  of  a  simple  and  popular  character,  and  designed 
for  the  general  public,  Professor  Schubert  has  incorporated  much  of  his  original 
research. 

Thomas  J.  McCormack. 
La  Salle,  111.,  December,  1898. 


op; 


629 


-    :\ 


Copyright  by 

The  Open  Court  Publishing  Co. 

1899 


TRANSLATOR'S  NOTE. 


'T^HE  mathematical  essays  and  recreations  in  this  volume  are  by  one  of  the  most 
"*■  successful  teachers  and  text-book  writers  of  Germany.  The  monistic  construc- 
tion of  arithmetic,  the  systematic  and  organic  development  of  all  its  consequences 
from  a  few  thoroughly  established  principles,  is  quite  foreign  to  the  general  run  of 
American  and  English  elementary  text-books,  and  the  first  three  essays  of  Professor 
Schubert  will,  therefore,  from  a  logical  and  esthetic  side,  be  full  of  suggestions  for 
elementary  mathematical  teachers  and  students,  as  well  as  for  non-mathematical 
readers.  For  the  actual  detailed  development  of  the  system  of  ari*bmetic  here 
sketched,  we  may  refer  the  reader  to  Professor  Schubert's  volume  Arithmctik 
utid  Algebra,  recently  published  in  the  Goschen-Sammlung  (Goschen,  Leipsic), — 
an  extraordinarily  cheap  series  containing  many  other  unique  and  valuable  text- 
books in  mathematics  and  the  sciences. 

The  remaining  essays  on  "Magic  Squares,"  "The  Fourth  Dimension,"  and 
"The  History  of  the  Squaring  of  the  Circle,"  will  be  found  to  be  the  most  com- 
plete generally  accessible  accounts  in  English,  and  to  have,  one  and  all,  a  distinct 
educational  and  ethical  lesson. 

In  all  these  essays,  which  are  of  a  simple  and  popular  character,  and  designed 
for  the  general  public.  Professor  Schubert  has  incorporated  much  of  his  original 
research. 

Thomas  J.  McCormack. 
La  Salle,  111.,  December,  1898. 


95629 


CONTENTS. 

PAGE 

Notion  and  Definition  of  Number i 

Monism  in  Arithmetic ^ 

On  the  Nature  of  Mathematical  Knowledge 27 

The  Magic  Square 39 

The  Fourth  Dimension 64 

The  Squaring  of  the  Circle "2 


NOTION  AND  DEFINITION  OF  NUMBER. 

MANY  essays  have  been  written  on  the  definition  of  number. 
But  most  of  them  contain  too  many  technical  expressions, 
both  philosophical  and  mathematical,  to  suit  the  non-mathemati- 
cian. The  clearest  idea  of  what  counting  and  numbers  mean  may 
be  gained  from  the  observation  of  children  and  of  nations  in  the 
childhood  of  civilisation.  When  children  count  or  add,  they  use 
either  their  fingers,  or  small  sticks  of  wood,  or  pebbles,  or  similar 
things,  which  they  adjoin  singly  to  the  things  to  be  counted  or 
otherwise  ordinally  associate  with  them.  As  we  know  from  history, 
the  Romans  and  Greeks  employed  their  fingers  when  they  counted 
or  added.  And  even  to-day  we  frequently  meet  with  people  to  whom 
the  use   of  the  fingers  is  absolutely  indispensable  for  computation. 

Still  better  proof  that  the  accurate  association  of  such  "other" 
things  with  the  things  to  be  counted  is  the  essential  element  of  nu- 
meration are  the  tales  of  travellers  in  Africa,  telling  us  how  African 
tribes  sometimes  inform  friendly  nations  of  the  number  of  the  enemies 
who  have  invaded  their  domain.  The  conveyance  of  the  informa- 
tion is  effected  not  by  messengers,  but  simply  by  placing  at  spots 
selected  for  the  purpose  a  number  of  stones  exactly  equal  to  the 
number  of  the  invaders.  No  one  will  deny  that  the  number  of  the 
tribe's  foes  is  thus  communicated,  even  though  no  name  exists  for 
this  number  in  the  languages  of  the  tribes.  The  reason  why  the 
fingers  are  so  universally  employed  as  a  means  of  numeration  is, 
that  every  one  possesses  a  definite  number  of  fingers,  sufficiently 
large  for  purposes  of  computation  and  that  they  are  always  at  hand. 

Besides  this  first  and  chief  element  of  numeration  which,  as  we 


2  NOTION  AND  DEFINITION  OF  NUiNIBER. 

have  seen,  is  the  exact,  individual  conjunction  or  association  of  other 
things  with  the  things  to  be  counted,  is  to  be  mentioned  a  second 
important  element,  which  in  some  respects  perhaps  is  not  so  abso- 
lutely essential;  namely,  that  the  things  to  be  counted  shall  be  re- 
garded as  of  the  same  kind.  Thus,  any  one  who  subjects  apples  and 
nuts  collectively  to  a  process  of  numeration  will  regard  them  for  the 
time  being  as  objects  of  the  same  kind,  perhaps  by  subsuming  them 
under  the  common  notion  of  fruit.  We  may  therefore  lay  down  pro- 
visionally the  following  as  a  definition  of  counting  :  to  count  a  group 
of  things  is  to  regard  the  things  as  the  same  in  kind  and  to  associate 
ordinally,  accurately,  and  singly  with  them  other  things.  In  writing, 
we  associate  with  the  things  to  be  counted  simple  signs,  like  points, 
strokes,  or  circles.  The  form  of  the  symbols  we  use  is  indifferent. 
Neither  need  they  be  uniform.  It  is  also  indifferent  what  the  spa- 
tial relations  or  dispositions  of  these  symbols  are.  Although,  of 
course,  it  is  much  more  convenient  and  simpler  to  fashion  symbols 
growing  out  of  operations  of  counting  on  principles  of  uniformity 
and  to  place  them  spatially  near  each  other.  In  this  manner  are 
produced  what  I  have  called  *  natural  number-pictures  ;  for  ex- 
ample. 


etc. 


Now-a-days  such  natural  number-pictures  are  rarely  employed,  and 
are  to  be  seen  only  on  dominoes,  dice,  and  sometimes,  also,  on  play- 
ing-cards. 

It  can  be  shown  by  archaeological  evidence  that  originally  nu- 
meral writing  was  made  up  wholly  of  natural  number-pictures.  For 
example,  the  Romans  in  early  times  represented  all  numbers,  which 
were  written  at  all,  by  assemblages  of  strokes.  We  have  remnants 
of  this  writing  in  the  first  three  numerals  of  the  modern  Roman  sys- 
tem. If  we  needed  additional  evidence  that  the  Romans  originall}' 
employed  natural  number-signs,  we  might  cite  the  passage  in  Livy, 
VII.  3,  where  we  are  told,  that,  in  accordance  with  a  very  ancient 
law,  a  nail  was  annually  driven  into  a  certain  spot  in  the  sanctuary  of 

*  System  dcr  Arithjjictik.     (Potsdam  :   Aug.  Stein.      1885.) 


NOTION  AND  DEFINITION  OF  NUMBER.  3, 

Minerva,  the  ' '  inventrix  "  of  counting,  for  the  purpose  of  showing  the 
number  of  years  which  had  elapsed  since  the  building  of  the  edifice. 
We  learn  from  the  same  source  that  also  in  the  temple  at  Volsinii 
nails  were  shown  which  the  Etruscans  had  placed  there  as  marks  for 
the  number  of  years. 

Also  recent  researches  in  the  civilisation  of  ancient  Mexico  show 
that  natural  number-pictures  were  the  first  stage  of  numeral  nota- 
tion. Whosoever  has  carefully  studied  in  any  large  ethnographical 
collection  the  monuments  of  ancient  Mexico,  will  surely  have  re- 
marked that  the  nations  which  inhabited  Mexico  before  its  conquest 
by  the  Spaniards,  possessed  natural  number-signs  for  all  numbers 
from  one  to  nineteen,  which  they  formed  by  combinations  of  circles. 
If  in  our  studies  of  the  past  of  modern  civilised  peoples,  we  meet 
with  natural  number-pictures  only  among  the  Greeks  or  Romans, 
and  some  Oriental  nations,  the  reason  is  that  the  other  nations,  as  the 
Germans,  before  they  came  into  contact  with  the  Romans  and  adopted 
the  more  highly  developed  notation  of  the  latter,  were  not  yet  suffi- 
ciently advanced  in  civilisation  to  feel  any  need  of  expressing  num- 
bers symbolically.  But  since  the  most  perfect  of  all  systems  of  nu- 
meration, the  Hindu  system  of  "local  value,"  was  introduced  and 
adopted  in  Europe  in  the  twelfth  century,  the  Roman  numeral  sys- 
tem gradually  disappeared,  at  least  from  practical  computation,  and 
at  present  we  are  only  reminded  by  the  Roman  characters  of  inscrip- 
tions of  the  first  and  primitive  stage  of  all  numeral  notation.  To- 
day we  see  natural  number-pictures,  except  in  the  above-mentioned 
games,  only  very  rarely,  as  where  the  tally-men  of  wharves  or  ware- 
houses make  single  strokes  with  a  pencil  or  a  piece  of  chalk,  one  for 
each  bale  or  sack  which  is  counted. 

As  in  writing  it  is  of  consequence  to  associate  with  each  of  the 
things  to  be  counted  some  simple  sign,  so  in  speaking  it  is  of  con- 
sequence to  utter  for  each  single  thing  counted  some  short  sound. 
It  is  quite  indifferent  here  what  this  sound  is  called,  also  whether 
the  sounds  which  are  associated  with  the  things  to  be  counted  are 
the  same  in  kind  or  not,  and  finally,  whether  they  are  uttered  at 
equal  or  unequal  intervals  of  time.  Yet  it  is  more  convenient  and 
simpler  to  employ  the  same  sound  and  to  observe  equal  intervals  in 


4  NOTION  AND  DEFINITION  OF  NUMBER. 

their  utterance.     We  arrive  thus  at  natural  number-words.     For  ex- 
ample, utterances  like, 

oh,  oh-oh,  oh-oh-oh,  oh-oh-oh-oh,  oh-oh-oh-oh-oh, 
are  natural  number-words  for  the  numbers  from  one  to  five.  Num- 
ber-words of  this  description  are  not  now  to  be  found  in  any  known 
language.  And  yet  we  hear  such  natural  number-words  constantly, 
every  day  and  night  of  our  lives  ;  the  only  difference  being  that  the 
speakers  are  not  human  beings  but  machines — namely,  the  striking- 
apparatus  of  our  clocks. 

Word-forms  of  the  kind  described  are  too  inconvenient,  how- 
ever, for  use  in  language,  not  only  for  the  speaker,  on  account  of 
their  ultimate  length,  but  also  for  the  hearer,  who  must  be  constantly 
on  the  qui  vive  lest  he  misunderstand  a  numeral  word  so  formed.  It 
has  thus  come  about  that  the  languages  of  men  from  time  imme- 
morial have  possessed  numeral  words  which  exhibit  no  trace  of  the 
original  idea  of  single  association.  But  if  we  should  always  select 
for  every  new  numeral  word  some  new  and  special  verbal  root,  we 
should  find  ourselves  in  possession  of  an  inordinately  large  number 
of  roots,  and  too  severely  tax  our  powers  of  memory.  Accordingly, 
the  languages  of  both  civilised  and  uncivilised  peoples  always  con- 
struct their  words  for  larger  numbers  from  words  for  smaller  num- 
bers. What  number  we  shall  begin  with  in  the  formation  of  com- 
pound numeral  words  is  quite  indifferent,  so  far  as  the  idea  of  num- 
ber itself  is  concerned.  Yet  we  find,  nevertheless,  in  nearly  all 
languages  one  and  the  same  number  taken  as  the  first  station  in  the 
formation  of  compound  numeral  words,  and  this  number  is  ten. 
Chinese  and  Latins,  Fins  and  Malays,  that  is,  peoples  who  have  no 
linguistic  relationship,  all  display  in  the  formation  of  numeral  words 
the  similarity  of  beginning  with  the  number  ten  the  formation  of 
compound  numerals.  No  other  reason  can  be  found  for  this  striking 
agreement  than  the  fact  that  all  the  forefathers  of  these  nations  pos- 
sessed ten  fingers. 

Granting  it  were  impossible  to  prove  in  any  other  way  that 
people  originally  used  their  fingers  in  reckoning,  the  conclusion 
could  be  inferred  with  sufficient  certainty  solely  from  this  agreement 
with  regard  to  the  first   resting-point  in  tlic  formation  of  compound 


NOTION  AND  DEFINITION  OF  NUMBER.  5 

numerals  among  the  most  various  races.  In  the  Indo-Germanic 
tongues  the  numeral  words  from  ten  to  ninety-nine  are  formed  by 
composition  from  smaller  numeral  words.  Two  methods  remain 
for  continuing  the  formation  of  the  numerals  :  either  to  take  a  new 
root  as  our  basis  of  composition  (hundred)  or  to  go  on  counting 
from  ninety-nine,  saying  tenty,  eleventy,  etc.  If  we  were  logically 
to  follow  out  this  second  method  we  should  get  tenty-ty  for  a  thou- 
sand, tenty-ty-ty  for  ten  thousand,  etc.  But  in  the  utterance  of  such 
words,  the  syllable  iy  would  be  so  frequently  repeated  that  the  same 
inconvenience  would  be  produced  as  above  in  our  individual  num- 
ber-pictures. For  this  reason  the  genius  which  controls  the  for- 
mation of  speech  took  the  first  course. 

But  this  course  is  only  logically  carried  out  in  the  old  Indian 
numeral  words.  In  Sanskrit  we  not  only  have  for  ten,  hundred, 
and  thousand  a  new  root,  but  new  bases  of  composition  also  exist 
for  ten  thousand,  one  hundred  thousand,  ten  millions,  etc.,  which 
are  in  no  wise  related  with  the  words  for  smaller  numbers.  Such 
roots  exist  among  the  Hindus  for  all  numerals  up  to  the  number  ex- 
pressed by  a  one  and  fifty-four  appended  naughts.  In  no  other  lan- 
guage do  we  find  this  principle  carried  so  far.  In  most  languages  the 
numeral  words  for  the  numbers  consisting  of  a  one  with  four  and 
five  appended  naughts  are  compounded,  and  in  further  formations 
use  is  made  of  the  words  million,  billion,  trillion,  etc.,  which  really 
exhibit  only  one  root,  before  which  numeral  words  of  the  Latin 
tongue  are  placed. 

Besides  numeral  word-systems  based  on  the  number  icn,  only  lo- 
gical systems  are  found  based  on  the  number  five  and  on  the  number 
twenty.  Systems  of  numeral  words  which  have  the  basis  five  occur 
in  equatorial  Africa.  (See  the  language-tables  of  Stanley's  books 
on  Africa.)  The  Aztecs  and  Mayas  of  ancient  Mexico  had  the  base 
twenty.  In  Europe  it  was  mainly  the  Celts  who  reckoned  with 
twenty  as  base.  The  French  language  still  shows  some  few  traces 
of  the  Celtic  vicenary  system,  as  in  its  word  for  eighty,  quatre-vitigt. 
The  choice  of  five  and  of  twenty  as  bases  is  explained  simply  enough 
by  the  fact  that  each  hand  has  five  fingers,  and  that  hands  and  feet 
together  have  twenty  fingers  and  toes. 


6  NOTION  AND  DEFINITION  OF  NUMBER. 

As  we  see,  the  languages  of  humanity  no'\V  no  longer  possess 
natural  number-signs  and  number- words,  but  employ  names  and 
systems  of  notation  adopted  subsequently  to  this  first  stage.  Ac- 
cordingly, we  must  add  to  the  definition  of  counting  above  given  a 
third  factor  or  element  which,  though  not  absolutely  necessary,  is 
yet  important,  namely,  that  we  must  be  able  to  express  the  results 
of  the  above-defined  associating  of  certain  other  things  with  the 
things  to  be  counted,  by  some  conventional  sign  or  numeral  word. 

Having  thus  established  what  counting  or  numbering  means, 
we  are  in  a  position  to  define  also  the  notion  of  number,  which  we 
do  by  simply  saying  that  by  number  we  understand  iJic  results  of 
counting  or  numeration.  These  are  naturally  composed  of  two  ele- 
ments. First,  of  the  ordinary  number-word  or  number-sign  ;  and 
secondly,  of  the  word  standing  for  the  specific  things  counted.  For 
example,  eight  men,  seven  trees,  five  cities.  When,  now,  we  have 
counted  one  group  of  things,  and  subsequently  also  counted  another 
group  of  things  of  the  same  kind,  and  thereupon  we  conceive  the 
tvv-o  groups  of  things  combined  into  a  single  group,  we  can  save 
ourselves  the  labor  of  counting  the  things  a  third  time  by  blending 
the  number-pictures  belonging  to  the  two  groups  into  a  single  num- 
ber-picture belonging  to  the  whole.  In  this  way  we  arrive  on  the 
one  hand  at  the  idea  of  addition,  and  on  t-iie  other,  at  the  notion  of 
"unnamed"  number.  Since  we  have  no  means  of  tellfng  from  the 
two  original  number-pictures  and  the  third  one  which  is  produced 
from  these,  the  kind  or  character  of  the  things  counted,  we  are  ulti- 
mately led  in  our  conception  of  number  to  abstract  wholly  from  the 
nature  of  the  things  counted,  and  to  form  the  definition  of  unnamed 
number. 

We  thus  see  that  to  ascend  from  the  notion  of  named  number 
to  the  notion  of  unnamed  number,  the  notion  of  addition,  joined  to 
a  high  power  of  abstraction,  is  necessary.  Here  again  our  theory 
is  best  verified  by  observations  of  children  learning  to  count  and 
add.  A  child,  in  beginning  arithmetic,  can  well  understand  what 
five  pens  or  five  chairs  are,  but  he  cannot  be  made  to  understand 
from  this  alone  what  five  abstractly  is.  But  if  we  put  beside  the 
first  five  pens  three  other  pens,  or  beside  the  five  chairs  three  other 


NOTION  AND  DEFINITION  OF  NUMBER.  7 

chairs,  we  can  usualJy  bring  the  child  to  see  that  five  things  plus 
three  things  are  always  eight  things,  no  matter  of  what  nature  the 
things  are,  and  that  accordingly  we  need  not  always  specify  in 
counting  what  kind  of  things  we  mean.  At  first  we  always  make 
the  answer  to  our  question  of  what  five  plus  three  is,  easy  for  the 
child,  by  relieving  him  of  the  process  of  abstraction,  which  is  neces- 
sary to  ascend  from  the  named  to  the  unnamed  number,  an  end 
which  we  accomplish  by  not  asking  first  what  five  plus  three  is,  but 
by  associating  with  the  numbers  words  designating  things  within 
the  sphere  of  the  child's  experience,  for  example,  by  asking  how 
many  five  pens  plus  three  pens  are. 

The  preceding  reflexions  have  led  us  to  the  notion  of  unnamed 
or  abstract  numbers.  The  arithmetician  calls  these  numbers  posi- 
tive whole  numbers,  or  positive  integers,  as  he  knows  of  other  kinds 
of  numbers,  for  example,  negative  numbers,  irrational  numbers,  etc. 
Still,  observation  of  the  world  of  actual  facts,  as  revealed  to  us  by  our 
senses,  can  naturally  lead  us  only  to  positive  whole  numbers,  such 
only,  and  no  others,  being  results  of  actual  counting.  All  other  kinds 
of  numbers  are  nothing  but  artificial  inventions  of  mathematicians 
created  for  the  purpose  of  giving  to  the  chief  tool  of  the  mathema- 
tician, namely,  arithmetical  notation,  a  more  convenient  and  more 
practical  form,  so  that  the  solution  of  the  problems  which  arise  in 
mathematics  may  be  simplified.  All  numbers,  excepting  the  results 
of  counting  above  defined,  are  and  remain  mere  symbols,  which, 
although  they  are  of  incalculable  value  in  mathematics,  and,  there- 
fore, can  scarcely  be  dispensed  with,  yet  could,  if  it  were  a  ques- 
tion of  principle,  be  avoided.  Kronecker  has  shown  that  any  prob- 
lem in  which  positive  whole  numbers  are  given,  and  only  such  are 
sought,  always  admits  of  solution  without  the  help  of  other  kinds  of 
numbers,  although  the  employment  of  the  latter  wonderfully  sim- 
plifies the  solution. 

How  these  derived  species  of  numbers,  by  the  logical  applica- 
tion of  a  single  principle,  flow  naturally  from  the  notion  of  number 
and  of  addition  above  deduced,  I  shall  show  in  the  next  article  en- 
titled "Monism  in  Arithmetic." 


MONISM  IN  ARITHMETIC. 

TN  HIS  Primer  of  Philosophy,  Dr.  Paul  Carus  defines  monism 
^  as  a  "unitary  conception  of  the  world."  Similarly,  we  shall 
understand  by  monism  in  a  science  the  unitary  conception  of  that 
science.  The  more  a  science  advances  the  more  does  monism  domi- 
nate it.  An  example  of  this  is  furnished  by  physics.  Whereas 
formerly  physics  was  made  up  of  wholly  isolated  branches,  like 
Mechanics,  Heat,  Optics,  Electricity,  and  so  forth,  each  of  which 
received  independent  explanations,  physics  has  now  donned  an  al- 
most absolute  monistic  form,  by  the  reduction  of  all  phenomena  to 
the  motions  of  molecules.  For  example,  optical  and  electrical  phe- 
nomena, we  now  know,  are  caused  by  the  undulatory  movements 
of  the  ether,  and  the  length  of  the  ether-waves  constitutes  the  sole 
difference  between  light  and  electricity. 

Still  more  distinctly  than  in  physics  is  the  monistic  element 
displayed  in  pure  arithmetic,  by  which  we  understand  the  theory  of 
the  combination  of  two  numbers  into  a  third  by  addition  and  the 
direct  and  indirect  operations  springing  out  of  addition.  Pure  arith- 
metic is  a  science  which  has  completely  attained  its  goal,  and  which 
can  prove  that  it  has,  exclusively  b}'  internal  evidence.  For  it  may 
be  shown  on  the  one  hand  that  besides  the  seven  familiar  operations 
of  addition,  subtraction,  multiplication,  division,  involution,  evolu- 
tion, and  the  finding  of  logarithms,  no  other  operations  are  defin- 
able which  present  anything  essentially  new;  and  on  the  other  hand 
that  fresh  extensions  of  the  domain  of  numbers  beyond  irrational, 
imaginary,  and  complex  numbers  are  arithmetically  impossible. 
Arithmetic  may  be  compared  to  a  tree  that  has  completed  its  growth. 


MONISM  IN  ARITHMETIC.  Q 

the  boughs  and  branches  of  which  maj^  still  increase  in  size  or  even 
give  forth  fresh  sprouts,  but  whose  main  trunk  has  attained  its  full- 
est development. 

Since  arithmetic  has  arrived  at  its  maturity,  the  more  profound 
investigation  of  the  nature  of  numbers  and  their  combinations  shows 
that  a  unitary  conception  of  arithmetic  is  not  only  possible  but  also 
necessary.  If  we  logically  abide  by  this  unitary  conception,  we  ar- 
rive, starting  from  the  notion  of  counting  and  the  allied  notion  of 
addition,  at  all  conceivable  operations  and  at  all  possible  extensions  of 
the  notion  of  number.  Although  previously  expressed  by  Grassmann, 
Hankel,  E.  Schroder,  and  Kronecker,  the  author  of  the  present  ar- 
ticle, in  his  "  System  of  Arithmetic,"  Potsdam,  1885,  was  the  first 
to  work  out  the  idea  referred  to,  fully  and  logically  and  in  a  form 
comprehensible  for  beginners.  This  book,  which  Kronecker  in  his 
"Notion  of  Number,"  an  essay  published  in  Zeller's  jubilee  work, 
makes  special  mention  of,  is  intended  for  persons  proposing  to  learn 
arithmetic.  As  that  cannot  be  the  object  of  the  readers  of  these  es- 
says, whose  purpose  will  rather  be  the  study  of  the  logical  construc- 
tion of  the  science  from  some  single  fundamental  principle,  the  fol- 
lowing pages  will  simply  give  of  the  notions  and  laws  of  arithmetic 
what  is  absolutely  necessary  for  an  understanding  of  its  develop- 
ment. 

The  starting-point  of  arithmetic  is  the  idea  of  counting  and  of 
number  as  the  result  of  counting.  On  this  subject,  the  reader  is  re- 
quested to  read  the  first  essay  of  this  collection.  It  is  there  shown 
that  the  idea  of  addition  springs  immediately  from  the  idea  of  count- 
ing. As  in  counting  it  is  indifferent  in  what  order  we  count,  so  in 
addition  it  is  indifferent,  for  the  sum,  or  the  result  of  the  addition, 
whether  we  add  the  first  number  to  the  second  or  the  second  to  the 
first.  This  law,  which  in  the  symbolic  language  of  arithmetic,  is 
expressed  by  the  formula 

a-\-  b^b  -\-  a, 

is  called  the  commutative  law  of  addition.  Notwithstanding  this  law, 
however,  it  is  evidently  desirable  to  distinguish  the  two  quantities 
which  are  to  be  summed,  and  out  of  which  the  sum  is  produced,  by 
special  names.      As  a  fact,  the  two  summands  usually  are  distin- 


lO  MONISM  IN  ARITHMETIC. 

guished  in  some  way,  for  example,  by  saying  a  is  to  be  increased  by 
b,  or  b  is  to  be  added  to  a,  and  so  forth.  Here,  it  is  plain,  a  is  al- 
ways something  that  is  to  be  increased,  b  the  increase.  Accordingly 
it  has  been  proposed  to  call  the  number  which  is  regarded  in  addi- 
tion as  the  passive  number  or  the  one  to  be  changed,  the  augend, 
and  the  other  which  plays  the  active  part,  which  accomplishes  the 
change,  so  to  speak,  the  incretnent.  Both  words  are  derived  from 
the  Latin  and  are  appropriately  chosen.  Augend  is  derived  from 
augere,  to  increase,  and  signifies  that  which  is  to  be  increased  ;  in- 
crement comes  from  increscere,  to  grow,  and  signifies  as  in  its  ordi- 
nary meaning  what  is  added. 

Besides  the  commutative  law  one  other  follows  from  the  idea  of 
counting — the  associative  lata  of  addition.  This  law,  which  has  ref- 
erence not  to  two  but  to  three  numbers,  states  that  having  a  certain 
sum,  a  -\-  b,  it  is  indifferent  for  the  result  whether  we  increase  the 
increment  b  of  that  sum  by  a  number,  or  whether  we  increase  the 
sum  itself  by  the  same  number.  Expressed  in  the  symbolic  lan- 
guage of  arithmetic  this  law  reads, 

a^{b^c')  =  {^a-Yb)^c. 
To  obtain  now  all  the  rules  of  addition  we  have  only  to  apply  the 
two  laws  of  commutation  and  association  above  stated,  though  fre- 
quently, in  the  deduction  of  the  same  rule,  each  must  be  applied 
many  times.  I  may  pass  over  here  both  the  rules  and  their  estab- 
lishment. 

In  addition,  two  numbers,  the  augend  a  and  the  increment  b 
are  combined  into  a  third  number  c,  the  sum.  From  this  operation 
spring  necessarily  two  inverse  operations,  the  common  feature  of 
which  is,  that  the  sum  sought  in  addition  is  regarded  in  both  as 
known,  and  the  difference  that  in  the  one  the  augend  also  is  regarded 
as  known,  and  in  the  other  the  increment.  If  we  ask  what  number 
added  to  a  gives  c,  we  seek  the  increment.  If  we  ask  what  number 
increased  by  b  gives  c,  we  seek  the  augend.  As  a  matter  of  reckon- 
ing, the  solution  of  the  two  questions  is  the  same,  since  by  the  com- 
mutative law  of  addition  aA^b^^bA^a.  Consequently,  only  one 
common  name  is  in  vise  for  the  two  inverses  of  addition,  namely, 
subtract  ion.    But  with  respect  to  the  notions  involved,  the  two  oper- 


MONISM  IN  ARITHMETIC.  II 

ations  do  differ,  and  it  is  accordingly  desirable  in  a  logical  investi- 
gation of  the  structure  of  arithmetic,  to  distinguish  the  two  by  dif- 
ferent names.  As  in  all  probability  no  terms  have  yet  been  sug- 
gested for  these  two  kinds  of  subtraction,  I  propose  here  for  the 
first  time  the  following  words  for  the  two  operations,  namely,  de- 
tract io7i  to  denote  the  finding  of  the  increment,  and  suhlertraction  to 
denote  the  finding  of  the  augend.  We  obtain  these  terms  simply 
enough  by  thinking  of  the  augmentation  of  some  object  already  ex- 
isting. For  example,  the  cathedral  at  Cologne  had  in  its  tower  an 
augend  that  waited  centuries  for  its  increment,  which  was  only 
supplied  a  few  decades  ago.  As  the  cathedral  had  originally  a 
height  of  one  hundred  and  thirty  metres,  but  after  completion  was 
increased  in  height  twenty-six  metres,  of  the  total  height  of  one 
hundred  and  fifty-six  metres  one  hundred  and  thirty  metres  is  clearly 
the  augend  and  twent3'-six  metres  the  increment.  If,  now,  we  wished 
to  recover  the  augend  we  should  have  to  pull  down  (Latin,  dctrahere) 
the  upper  part  along  the  whole  height.  Accordingly,  the  finding  of 
the  augend  is  called  detract ioii.  If  we  sought  the  increment,  we 
should  have  to  pull  out  the  original  part  from  beneath  (Latin,  snbter- 
trahere).  For  this  reason,  the  finding  of  the  increment  is  called  sub- 
tertraction.  Owing  to  the  commutative  law,  the  two  inverse  opera- 
tions, as  matters  of  computation,  become  one,  which  bears  the  name 
of  subtraction.  The  sign  of  this  operation  is  the  minus  sign,  a  hori- 
zontal stroke.  The  number  which  originally  was  sum,  is  called  in 
subtraction  minuend  ;  the  number  which  in  addition  was  increment 
is  now  called  detractor;  the  number  which  in  addition  was  augend 
is  now  called  subtertractor.  Comprising  the  two  conceptually  dif- 
ferent operations  in  one  single  operation,  subtraction,  we  employ 
for  the  number  which  before  was  increment  or  augend,  the  term  sub- 
trahend, a  word  which  on  account  of  its  passive  ending  is  not  very 
good,  and  for  which,  accordingl}^  E.  Schroder  proposes  to  substi- 
tute the  word  subtrahcnt,  having  an  active  ending.  The  result  of 
subtraction,  or  what  is  the  same  thing,  the  number  sought,  is  called 
the  difference.     The  definition-formula  of  subtraction  reads 

a  —  b  -^  b  ^=  a, 
that  is,  a  minus  b  is  the  number  which  increased  by  b  gives  a,  or 


12  MONISM  IN  arithmetic: 

the  number  which  added  to  b  gives  a,  according  as  the  one  or  the 
other  of  the  two  operations  inverse  to  addition  is  meant.  From  the 
formula  for  subtraction,  and  from  the  rules  which  hold  for  addition, 
follow  now  at  once  the  rules  which  refer  to  both  addition  and  sub- 
traction.     These  rules  we  here  omit. 

From  the  foregoing  it  is  plain  that  the  minuend  is  necessarily- 
larger  than  the  subtrahent.  For  in  the  process  of  addition  the  minu- 
end was  the  sum,  and  the  sum  grew  out  of  the  union  of  tivo  natural 
number-pictures.*  Thus  5  minus  g,  or  1 1  minus  12,  or  8  minus  8, 
are  combinations  of  numbers  lulioUy  destitute  of  inea)2ing ;  for  no 
number,  that  is,  no  result  of  counting,  exists  that  added  to  9  gives 
the  sum  5,  or  added  to  12  gives  the  sum  11,  or  added  to  8  gives  8. 
What,  then,  is  to  be  done?  Shall  we  banish  entirely  from  arith- 
metic such  meaningless  combinations  of  numbers  ;  or,  since  they 
have  no  meaning,  shall  we  rather  invest  them  with  one?  If  we  do 
the  first,  arithmetic  will  still  be  confined  in  the  strait-jacket  into 
which  it  was  forced  by  the  original  definition  of  number  as  the  re- 
sult of  counting.  If  we  adopt  the  latter  alternative  we  are  forced 
to  extend  our  notion  of  number.  But  in  doing  this,  we  sow  the 
first  seeds  of  the  science  of  pure  arithmetic,  an  organic  body  of 
knowledge  which  fructifies  all  other  provinces  of  science. 

What  significance,  then,  shall  we  impart  to  the  symbol 

5-9? 
Since  5  minus  9  possesses  no  significance  whatever,  we  may,  of 
course,  impart  to  it  an}^  significance  we  wish.  But  as  a  matter 
of  practical  convenience  it  should  be  invested  with  no  meaning 
that  is  likely  to  render  it  subject  to  exceptions.  As  the  form  of  the 
symbol  5  —  9  is  the  form  of  a  difference,  it  will  be  obviously  con- 
venient to  give  it  a  meaning  which  will  allow  us  to  reckon  with  it  as 
we  reckon  with  every  other  real  difference,  that  is,  with  a  difference 
in  which  the  minuend  is  larger  than  the  subtrahent.  This  being 
agreed  upon,  it  follows  at  once  that  all  such  symbols  in  which  the 
number  before  the  minus  sign  is  less  than  the  number  behind  it  by 
the  same  amount  may  be  put  equal  to  one  another.      It  is  practical, 

*  See  page  2,  supra. 


MONISM   IN  ARITHMETIC.  I3 

therefore,  to  comprise  all  these  symbols  under  some  one  single  sym- 
bol, and  to  construct  this  latter  symbol  so  that  it  will  appear  un- 
equivocally from  it  by  how  much  the  number  before  the  minus  sign 
is  less  than  the  number  behind  it.  This  difference,  accordingl}^  is 
written  down  and  the  minus  sign  placed  before  it. 

If  the  two  numbers  of  such  a  differential /(vv//  are  equal,  a  totally 
new  sign  must  be  invented  for  the  expression  of  the  fact,  having 
no  relation  to  the  signs  which  state  results  of  counting.  This  in- 
vention was  not  made  by  the  ancient  Greeks,  as  one  might  naturally 
suppose  from  the  high  mathematical  attainments  of  that  people,  but 
by  Hindu  Brahman  priests  at  the  end  of  the  fourth  century  after 
Christ.  The  symbol  which  they  invented  they  called  fsiphra,  empty, 
whence  is  derived  the  English  ciplicr.  The  form  of  this  sign  has  been 
different  in  different  times  and  with  different  peoples.  But  for  the 
last  two  or  three  centuries,  since  the  symbolic  language  of  arith- 
metic has  become  thoroughly  established  as  an  international  char- 
acter, the  form  of  the  sign  has  been  0  (French  zero,  German  nuW). 

In  calling  this  symbol  and  the  symbols  formed  of  a  minus  sign 
followed  by  a  result  of  counting,  numbers,  we  widen  the  province  of 
numbers,  which  before  was  wholly  limited  to  results  of  counting. 
In  no  other  way  can  zero  and  the  negative  numbers  be  introduced 
into  arithmetic.  No  man  can  prove  that  7  minus  1 1  is  equal  to  i 
minus  5.  Originally,  both  are  meaningless  symbols.  And  not  until 
we  agree  to  impart  to  them  a  significance  which  allows  us  to  reckon 
with  them  as  we  reckon  with  real  differences  are  we  led  to  a  state- 
ment of  identity  between  7  minus  11  and  i  minus  5.  It  was  a  long 
time  before  the  negative  numbers  mentioned  acquired  the  full  rights 
of  citizenship  in  arithmetic.  Cardan  called  them,  in  his  Ars  Magna, 
1545,  ////■/;/d'/7_/f(f// (imaginary  numbers),  as  distinguished  from  numei-i 
vcri  (real  numbers).  Not  until  Descartes,  in  the  first  half  of  the 
seventeenth  century,  was  any  one  bold  enough  to  substitute  niimeri 
ficti  and  numcri  veri  indiscriminately  for  the  same  letter  of  algebraic, 
expressions. 

We  have  invested,  thus,  combinations  of  signs  originally  mean- 
ingless, in  which  a  smaller  number  stood  before  than  after  a  minus 
sign,  with  a  meaning  which  enables  us  to  reckon  with  such  apparent 


14  MONISM  IN  ARITHMETIC. 

differences  exactly  as  we  do  with  ordinary  differences.  Now  it  is 
just  this  practical  shift  of  imparting  meanings  to  combinations,  which 
logically  applied  deduces  naturally  the  whole  system  of  arithmetic 
from  the  idea  of  counting  and  of  addition,  and  which  we  may  char- 
acterise, therefore,  as  the  foundation-priuciplc  of  its  whole  construc- 
tion. This  principle,  which  Hankel  once  called  the  principle  of  pcr- 
7nanence,  but  which  I  prefer  to  call  the  principle  of  no  exception, 
may  be  stated  in  general  terms  as  follows  : 

In  the  co7istruction  of  aritlimetic  every  combination  of  two  previously 
defined  mnnbers  by  a  sign  for  a  previously  defined  operation  (^plus,  minus, 
times,  etc. )  shall  be  invested  with  nicaniiig,  even  where  the  original  defi- 
nition of  the  operation  used  excludes  such  a  combination;  and  the  mean- 
ing imparted  is  to  be  such  that  the  combination  considered  shall  obey  the 
same  formula  of  definition  as  a  co7nbination  having  from  the  outset  a  sig- 
nification, so  that  the  old  laws  of  reckoni?ig  shall  still  hold  good  and  may 
still  be  applied  to  it. 

A  person  who  is  competent  to  apply  this  principle  rigorously 
and  logically  will  arrive  at  combinations  of  numbers  whose  results 
are  termed  irrational  or  imaginary  with  the  same  necessity  and  fa- 
cility as  at  the  combinations  above  discussed,  whose  results  are 
termed  negative  numbers  and  zero.  To  think  of  such  combinations 
as  results  and  to  call  the  products  reached  also  "  numbers  "  is  a  mis- 
use of  language.  It  were  better  if  we  used  the  phrase  /^vwi-  of  num- 
bers for  all  numbers  that  are  not  the  results  of  counting.  But  usus 
tyr  annus! 

It  will  now  be  my  task  to  show  how  all  numbers  at  which  arith- 
metic ever  has  arrived  or  ever  can  arrive  naturally  flow  from  the 
simple  application  6i  the  principle  of  no  exception. 

Owing  to  the  commutative  and  associative  laws  for  addition  it 
is  wholly  indifferent  for  the  result  of  a  series  of  additive  processes 
in  what  order  the  numbers  to  be  summed  are  added.      For  example, 

«+(/,  +  ,.  +  ,/)  ^^e-Yf)^{^a^b^c~)-^  (./+  e)  -f /. 
The  necessary  consequence  of  this  is  that  we  may  neglect  the  con- 
sideration of  the  order  of   the   numbers  and   give  heed  only  to  what 
the  quantities  are  that  are  to  be  summed,  and,  when  they  are  equal, 
take  note  of  onl\-  two  things,  namely,  of  what  the  quantit}-  which  is 


MONISM  IN  ARITHMETIC.  I5 

to  be  repeatedly  summed  is  called  and  how  ofte«  it  occurs.  We 
thus  reach  the  notion  of  multiplication.  To  multiply  ahy  b  means 
to  form  the  sum  of  I?  numbers  each  of  which  is  called  a.  The  num- 
ber conceived  summed  is  called  the  multiplicand,  the  number  which 
indicates  or  counts  how  often  the  first  is  conceived  summed  is  called 
the  multiplier. 

It  appears  hence,  that  the  multiplier  must  be  a  result  of  count- 
ing, or  a  number  in  the  original  sense  of  the  word,  but  that  the  mul- 
tiplicand may  be  any  number  hitherto  defined,  that  is,  may  also  be 
zero  or  negative.  It  also  follows  from  this  definition  that  though 
the  multiplicand  may  be  a  concrete  number  the  multiplier  cannot. 
Therefore,  the  commutative  law  of  multiplication  does  not  hold 
when  the  multiplicand  is  concrete.  For,  to  take  an  example,  though 
there  is  sense  in  requiring  four  trees  to  be  summed  three  times, 
there  is  no  sense  in  conceiving  the  number  three  summed  "four 
trees  times."  When,  however,  multiplicand  and  multiplier  are  un- 
named results  of  counting,  (abstract  numbers,)  two  fundamental 
laws  hold  in  multiplication,  exactly  analogous  to  the  fundamental 
laws  of  addition,  namely,  the  law  of  commutation  and  the  law  of 
association.      Thus,  • 

a  times  b^^b  tijjies  a, 
and,  a  times  {J)  times  e)  =  {a  times  b)  times  c. 
The  truth  and  correctness  of  these  laws  will  be  evident,  if  keeping 
to  the  definition  of  multiplication  as  an  abbreviated  addition  of  equal 
summands,  we  go  back  to  the  laws  of  addition.  Owing  to  the  com- 
mutative law  it  is  unnecessary,  for  purposes  of  practical  reckoning, 
to  distinguish  multiplicand  and  multiplier.  Both  have,  therefore,  a 
common  name  :  factor.  The  result  of  the  multiplication  is  called  the 
product;  the  symbol  of  multiplication  is  a  dot  (. )  or  a  cross  (X)j 
which  is  readT  "  times. "  Joined  with  the  fundamental  formula  above 
written  are  a  group  of  subsidiary  formulae  which  give  directions  how 
a  sum  or  difference  is  multiplied  and  how  multiplication  is  performed 
with  a  sum  or  difference.  I  need  not  enter,  however,  into  any  dis- 
cussion of  these  rules  here. 

As  the  combination  of  two  numbers  by  a  sign  of  multiplication 
has   no  significance  according  to  our  definition  of  multiplication, 


l6  MONISM  IX  ARITHMETIC. 

when  the  multipher  is  zero  or  a  negative  number,  it  will  be  seen 
that  we  are  again  in  a  position  where  it  is  necessary  to  apply  the 
above  explained  principle  of  no  exception.  We  revert,  therefore,  to 
what  we  above  established,  that  zero  and  negative  numbers  are  sym- 
bols which  have  the  form  of  differences,  and  lay  down  the  rule  that 
multiplications  with  zero  and  negative  numbers  shall  be  performed 
exactly  as  with  real  differences.  Why,  then,  is  minus  one  times 
minus  one,  for  example,  equal  to  plus  one?  For  no  other  reason 
than  that  minus  one  can  be  multiplied  with  an  ordinary  difference, 
as,  for  example,  8  minus  5,  by  first  multiplying  by  8,  then  multiply- 
ing by  5,  and  subtracting  the  differences  obtained,  and  because 
agreeably  to  the  principle  of  no  exception  we  must  say  that  the  mul- 
tiplication must  be  performed  according  to  exactly  the  same  rule 
Vi^ith  a  symbol  which  has  the  form  of  a  difference  whose  minuend  is 
less  by  one  than  its  subtrahent. 

As  from  addition  two  inverse  operations,  detraction  and  subter- 
traction,  spring,  so  also  from  multiplication  two  inverse  operations 
must  proceed  which  differ  from  each  other  simply  in  the  respect  that 
in  the  one  the  multiplicand  is  sought  and  in  the  other  the  multiplier. 
As  matters  of  computation,  these  two  inverse  operations  coalesce 
in  a  single  operation,  namel}',  division,  owing  to  the  validity  of  the 
commutative  law  in  multiplication.  But  in  so  far  as  they  are  differ- 
ent ideas,  they  must  be  distinguished.  As  most  civilised  languages 
distinguish  the  two  inverse  processes  of  multiplication  in  the  case 
in  which  the  multiplicand  is  a  line,  we  will  adopt  for  arithmetic  a 
name  which  is  used  in  this  exception.      Let  us  take  this  example, 

4  yards  x  3  ^  1 2  yards. 
If  .twelve  yards  and  four  yards  are  given,  and  the  multiplier  3  is 
sought,  I  ask,  how  many  summands,  each  equal  to  four  3'ards,  give 
twelve  yards,  or,  what  is  the  same  thing,  how  often  I  can  lay  a 
length  of  four  yards  on  a  length  of  twelve  yards?  But  this  is  /ncasttr- 
I'/i^t:;:  Secondl}',  if  twelve  yards  and  the  number  3  are  given,  and  the 
multiplicand  four  yards  is  sought,  I  ask  what  summand  it  is  which 
taken  three  times  gives  twelve  yards,  or,  what  is  the  same  thing, 
what  part  I  shall  obtain  if  I  cut  up  twelve  yards  into  three  equal 
parts?     But  this  is  partition,  or  parting.      If,  therefore,  the   multi- 


MONISM  IN  ARITHMETIC.  '  17 

plier  is  sought  we  call  the  division  nwasuriug,  and  if  the  multipli- 
cand is  sought,  we  call  it  parting.  In  both  cases  the  number  which 
was  originally  the  product  is  called  the  dividend,  and  the  result  the 
quotient.  The  number  which  originally  was  multiplicand  is  called 
the  measure ;  the  number  which  originally  was  multiplier  is  called 
the  parter.  The  common  name  for  measure  and  parter  is  divisor. 
The  common  symbol  for  both  kinds  of  division  is  a  colon,  a  hori- 
zontal stroke,  or  a  combination  of  both.  Its  definitional  formula 
reads, 

(a  -f-  <5)  .  ^  =  a,  or,       .  ^  =r  a. 

Accordingly,  dividing  a  by  />  means,  to  find  the  number  which  mul- 
tiplied by  I)  gives  a,  or  to  find  the  number  7vith  which  l>  must  be 
multiplied  to  produce  a.  From  this  formula,  together  with  the 
formulae  relative  to  multiplication,  the  well-known  rules  of  division 
are  derived,  which  I  here  pass  over. 

In  the  dividend  of  a  quotient  only  such  numbers  can  have  a 
place  which  are  the  product  of  the  divisor  with  some  previously  de- 
fined number.  For  example,  if  the  divisor  is  5  the  dividend  can 
only  be  5,  10,  15,  and  so  forth,  and  o,  — 5,  — 10  and  so  forth.  Ac- 
cordingly, a  stroke  of  division  having  underneath  it  5  and  above  it 
a  number  different  from  the  numbers  just  named  is  a  combination 
of  symbols  having  no  meaning.  For  example,  |  or  -^2.  are  meaning- 
less symbols.  Now,  conformably  to  the  principle  of  no  exception 
we  must  invest  such  symbols  which  have  the  form  of  a  quotient 
without  their  dividend  being  the  product  of  the  divisor  with  any 
number  yet  defined,  with  a  meaning  such  that  we  shall  be  able  to 
reckon  with  such  apparent  quotients  as  with  ordinary  quotients. 
This  is  done  by  our  agreeing  always  to  put  the  product  of  such  a 
quotient  form  with  its  divisor  equal  to  its  dividend.  In  this  way  we 
reach  the  definition  of  broken  numbers  ox  fractions,  which  by  the 
application  of  the  principle  of  no  exception  spring  from  division  ex- 
actly as  zero  and  negative  numbers  sprang  from  subtraction.  The 
latter  had  their  origin  in  the  impossibility  of  the  subtraction  ;  the 
former  have  their  origin  in  the  impossibility  of  the  division.    Putting 


l8  MONISM  IN  ARITHMETIC. 

together  now  both  these  extensions  of  the  domain  of  numbers,  we 
arrive  at  negative  fractional  nianbcrs. 

We  pass  over  the  easily  deduced  rules  of  computation  for  frac- 
tions and  shall  only  direct  the  reader's  attention  to  the  connexion 
which  exists  between  fractional  and  non-fractional  or,  as  we  usually 
say,  whole  numbers.  Since  the  number  12  lies  between  the  num- 
bers 10  and  15,  or,  what  is  the  same  thing,  10  <;  12  <;  15,  and  since 
10  :5  =  2,  15:5  =  3,  we  say  also  that  12  :  5  lies  between  2  and  3,  or 
that 

2<-V-<3- 

In  itself,  the  notion  of  "less  than  "  has  significance  only  for  results 
of  counting.  Consequently,  it  must  first  be  stated  what  is  meant 
when  it  is  said  that,2  is  less  than  ^J^.  Plainly,  nothing  is  meant  by 
this  except  that  2  times  5  is  less  than  12.  We  thus  see  that  every 
broken  number  can  be  so  interpolated  between  two  whole  numbers 
differing  from  each  other  only  by  i  that  the  one  shall  be  smaller 
and  the  other  greater,  where  smaller  and  greater  have  the  meaning 
above  given. 

From  the  above  definitions  and  the  laws  of  commutation  and 
association  all  possible  rules  of  computation  follow,  which  in  virtue 
of  the  principle  of  no  exception  now  hold  indiscriminately  for  all 
numbers  hitherto  defined.  It  is  a  consequence  of  these  rules,  again, 
that  the  combination  of  two  such  numbers  by  means  of  any  of  the 
operations  defined  must  in  every  case  lead  to  a  number  which  has 
been  already  defined,  that  is,  to  a  positive  or  negative  whole  or  frac- 
tional number,  or  to  zero.  The  sole  exception  is  the  case  where 
such  a  number  is  to  be  divided  by  zero.  If  the  dividend  also  is 
zero,  tliat  is,  if  we  have  the  combination  {},  the  expression  is  one 
which  stands  for  any  number  whatsoever,  because  any  number  what- 
soever, no  matter  what  it  is,  if  multiplied  by  zero  gives  zero.  But 
if  the  dividend  is  not  zero  but  some  other  number  a,  be  it  what  it 
will,  we  get  a  quotient  form  to  which  no  number  hitherto  defined 
can  be  equated.  But  we  discover  that  if  we  apply  the  ordinary  arith- 
metical rules  to  rt;^-  0  all  such  forms  may  be  equated  to  one  another 
both  when  a  is  positive  and  also  when  a  is  negative.  We  may  there- 
fore invent  two  new  signs  for  such  quotient  forms,  namely  -|-  oc  and 


MONISM  IN  ARITHMETIC.  ig 

—  oo.  We  find,  further,  that  in  transferring  the  notions  greater  and 
less  to  these  symbols,  -|~  °o  is  greater  than  any  positive  number, 
however  great,  and  — oo  is  smaller  than  any  negative  number,  how- 
ever small.  We  read  these  new  signs,  accordingly,  "plus  infinitely 
great"  and  "minus  infinitely  great." 

But  even  here  arithmetic  has  not  reached  its  completion,  al- 
though the  combination  of  as  many  previously  defined  numbers  as 
we  please  by  as  many  previously  defined  operations  as  we  please 
will  still  lead  necessarily  to  some  previously  defined  number.  Every 
science  must  make  every  possible  advance,  and  still  one  step  in  ad- 
vance is  possible  in  arithmetic.  For  in  virtue  of  the  laws  of  com- 
mutation and  association,  which  also  fortunately  obtain  in  multipli- 
cation, just  as  we  advance  from  addition  to  multiplication,  so  here 
again  we  may  ascend  from  multiplication  to  aii  operation  of  the  third 
degree.  For,  just  as  for  a -|- a -j- a -f- <z  we  read  \.a,  so  with  the  same 
reason  we  may  introduce  some  more  abbreviated  designation  for 
a. a. a. a.  The  introduction  of  this  new  operation  is  in  itself  simply 
a  matter  of  convenience  and  not  an  extension  of  the  ideas  of  arith- 
metic. But  if  after  having  introduced  this  operation  we  repeatedly 
apply  the  monistic  principle  of  arithmetic,  the  principle  of  no  ex- 
ception, we  reach  new  means  of  computation  which  have  led  to  un- 
dreamt of  advances  not  only  in  the  hands  of  mathematicians  but 
also  in  the  hands  of  natural  scientists.  The  abbreviated  designation 
mentioned,  which,  fructified  by  the  principle  of  no  exception,  can 
render  science  such  incalculable  services,  is  simply  that  of  writing 
for  a  product  of  b  factors  of  which  each  is  called  a,  a'',  which  we 
read  a  to  the  I?"'  power.  Here  a,  new  direct  operation,  that  of  invo- 
lution,  is  defined,  and  from  now  on  we  are  justified  in  distinguishing 
operations  which  are  not  inverses  of  others,  as  addition,  multiplica- 
tion, and  involution,  by  numbers  of  degree.  Addition  is  the  direct 
operation  of  the  first  degree,  multiplication  that  of  the  second  de- 
gree, and  involution  that  of  the  third  degree.  In  the  expression 
a''  the  passive  number  a  is  called  the  base,  the  active  number  b  the 
cxponefit,  the  result,  the  power. 

But  whilst  in  the  direct  operations  of  the  first  and  second  de- 
gree, the  laws  of  commutation  and  association  hold,  here  in  involu- 


20  MONISM   IN  ARITHMETIC. 

tion,  the  operation  of  third  degree,  the  two  laws  are  inappHcable, 
and  the  result  of  their  inapplicability  is  that  operations  of  a  still 
higher  degree  than  the  third  form  no  possible  advancement  of  pure 
arithmetic.  The  product  of  b  factors  a  is  not  equal  to  the  product 
of  a  factors  b;  that  is,  the  law  of  commutation  does  not  hold.  The 
only  two  different  integers  for  which  a  to  the  b'^'  power  is  equal  to  b 
to  the  a"'  power  are  2  and  4,  for  2  to  the  4"'  power  is  16,  and  4  to 
the  second  power  also  is  16.  So,  too,  the  law  of  association  as  a 
general  rule  does  not  hold.  For  it  is  hardly  the  same  thing  whether 
we  take  the  (//)'''  power  of  a  or  the  c'''  power  of  a''. 

From  the  definition  of  involution  follow  the  usual  rules  for  reck- 
oning with  powers,  of  which  we  shall  only  mention  one,  namely, 
that  the  (/--  —  c)"'  power  of  a  is  equal  to  the  result  of  the  division  of 
a  to  the  b"'  power  by  a  to  the  c"'  power.  If  we  put  here  c  equal  to 
b,  we  are  obliged,  by  the  principle  of  no  exception,  to  put  a  to  the 
0"'  power  equal  to  i;  a  new  result  not  contained  in  the  original  no- 
tion of  involution,  for  that  implied  necessarily  that  the  exponent 
should  be  a  result  of  counting.  Again,  if  we  make  b  smaller  than  c 
we  obtain  a  negative  exponent,  which  we  should  not  know  how  to 
dispose  of  if  we  did  not  follow  our  monistic  law  of  arithmetic.  Ac- 
cording to  the  latter,  a  to  the  {I)  —  e')"'  power  must  still  remain  equal 
to  a*  divided  by  a"  even  when  b  is  smaller  than  e.  Whence  follows 
that  a  to  the  minus  d'"  power  is  equal  to  i  divided  by  a  to  the 
d"'  power,  or  to  take  specific  numbers,  that  3  to  the  minus  1'"^  power 
is  equal  to  ^. 

At  this  point,  perhaps,  the  reader  will  inquire  what  a  raised  to  a 
fractional  power  is.  But  this  can  be  explained  only  when  we  have 
discussed  the  inverse  processes  of  involution,  to  which  we  now  pass. 

If  a^^=c,  we  may  ask  two  questions  :  first,  what  the  base  is 
which  raised  to  the  b"'  power  gives  e;  the  second,  what  the  exponent 
of  the  power  is  to  which  a  must  be  raised  to  produce  e.  In  the  first 
case  we  seek  the  base,  and  term  the  operation  which  yields  this  re- 
sult evolution;  in  the  second  case  we  seek  the  exponent  and  call  the 
operation  which  yields  this  exponent,  the  finding  of  the  logarithm. 
In  the  first  case,  we  write  \/ c^=a  (which  we  read,  the  b"'  root  of  e  is 
equal  to  a),  and  call  c  the  radieand,  b  the  exponent  of  the  root,  and  a 


MONISM  IN  ARITHMETIC.  21 

the  root.  In  the  second  case,  we  write  log„^  =  /^  (which  we  read,  the 
logarithm  of  c  to  the  base  a  is  equal  to  b'),  and  call  c  the  logarithm  and 
or  mimbcr,  a  the  base  of  the  logarithm,  and  b  the  logarithm. 

While,  owing- to  the  validity  of  the  law  of  commutation  in  addi- 
tion and  multiplication,  the  two  inverse  processes  of  those  opera- 
tions are  identical  so  far  as  computation  is  concerned,  here  in  the 
case  of  involution  the  two  inverse  operations  are  in  this  regard  es- 
sentially different,  for  in  this  case  the  law  of  commutation  does  not 
hold. 

From  the  definitional  formulae  for  evolution  and  the  finding  of 
logarithms,  namely, 

{y7y  =  c,  and  {a)  '°g«^  =  c, 
follow,  by  the  application  of  the  laws  of  involution,  the  rules  for 
computation  with  roots  and  logarithms.  These  rules  we  pass  over 
here,  only  remarking,  first,  that  for  the  present  {/c  has  meaning 
only  when  c  is  the  //*  power  of  some  number  already  defined  ;  and, 
secondly,  that  for  the  present  also  log^^r  has  meaning  only  when  c 
can  be  produced  by  raising  the  number  a  to  some  power  which  is  a 
number  already  defined.  In  the  phrase  "has  only  meaning  for  the 
present"  is  contained  a  possibility  of  new  extensions  of  the  domain 
of  number.  But  before  we  pass  to  those  extensions  we  shall  first 
make  use  of  the  idea  of  evolution  just  defined  to  extend  the  notion 
of  power  also  to  cases  in  which  the  exponent  is  a  fractional  number. 

According  to  the  original  definition  of  involution,  «*  was  mean- 
ingless except  where  b  was  a  result  of  counting.  But  afterwards, 
even  powers  which  had  for  their  exponents  zero  or  a  negative  integer 
could  be  invested  with  meaning.  Now  we  have  to  consider  the 
arithmetical  combination  "  a  raised  to  the  fractional  power  — ."  The 
principle  of  no  exception  compels  us  to  give  to  the  arithmetical  com- 
bination "  <z  to  the  -^-"'  power  "  a  significance  such  that  all  the  rules 
of  computation  will  hold  with  respect  to  it.  Now,  one  rule  that 
holds  is,  that  the  m"'  power  of  the  >i"'  power  of  a  is  equal  to  the 
{inY.n)"'  power  of  a.  Consequently,  the  q"'  power  of  a  raised  to  the 
— '*  power  must  be  equal  to  a  raised  to  a  power  whose  exponent  is 
equal  to  ^^^  times  q.  But  the  last-mentioned  product  gives,  according 
to  the  definition  of  division,  the  number  p.     Consequently  the  sym- 


22  MONISM  IN  ARITHMETIC. 

bol  a  to  the  -—"^  power  is  so  constituted  that  its  q^''  poAver  is  equal  to 
a  to  the/'''  power;  i.  e.,  it  is  equal  to  the  q"'  root  of  a^.  Similarly, 
we  find  that  the  symbol  "■a  to  the  minus  -^-"'  power"  must  be  put 
equal  to  i  divided  by  the  q'''  root  of  a  to  the  p"'  power,  if  we  are  to 
reckon  with  this  symbol  as  we  do  with  real  powers.  Again,  just  as 
a  to  the  b"'  power  is  invested  with  meaning  when  ^  is  a  fractional 
number,  so  some  meaning  harmonious  with  the  principle  of  no  ex- 
ception must  be  imparted  to  the  /'•''  root  of  c  where  b  is  a  positive  or 
negative  fractional  number.  For  example,  the  three-fourths"'  root 
of  8  is  equal  to  8  to  the  a  power,  that  is,  to  the  cube  root  of  8  to  the 
4''''  power,  or  16. 

The  principle  underlying  arithmetic  now  also  compels  us  to 
give  to  the  symbol  the  '■'■b"'  root  of  r"  a  meaning  when  c  is  not  the 
b"'  power  of  any  number  yet  defined.  First,  let  c  be  any  positive 
integer  or  fraction.  Then  always  to  be  able  to  reckon  with  the 
b"'  root  of  c  in  the  same  way  that  we  do  with  extractible  roots,  we 
must  agree  always  to  put  the  b"'  power  of  the  b"'  root  of  c  equal  to 
c — for  example,  (f  3)-  always  exactly  equal  to  3.  A  careful  inspec- 
tion of  the  new  symbols,  which  we  will  also  call  numbers,  shows,  that 
though  no  one  of  them  is  exactly  equal  to  a  number  hitherto  defined, 
yet  by  a  certain  extension  of  the  notions  greater  and  less,  two  num- 
bers of  the  character  of  numbers  already  defined  may  be  found  for 
each  such  new  number,  such  that  the  new  number  is  greater  than  the 
one  and  less  than  the  other  of  the  two,  and  that  further,  these  two 
numbers  may  be  made  to  differ  from  each  other  by  as  small  a  quan- 
tity as  we  please.      For  example, 

/'7^3  343  993     ,-^  3  <--  .^3  2  7  (^\Z 

The  number  3,  as  we  see,  is  here  included  between  two  limits  which 
are  the  third  powers  of  two  numbers  ^  and  |  whose  difference  is  -Jj^. 
We  could  also  have  arranged  it  so  that  the  difference  should  be 
equal  to  -j-J^^,  or  to  any  specified  number,  however  small.  Now,  in- 
stead of  putting  the  symbol  "  less  than  "  between  (|)^  and  3,  and 
between  3  and  (l)"^,  let  us  put  it  between  their  third  roots ;  for  ex- 
ample, let  us  say : 

^  <  ^^'^'<  f ,  meaning  by  this  that  (If  <  3  <  (f)3. 
In  this  sense  we  may  say  that  the  new  numbers  always  lie  between 


i  .  " 


MONISM  IN  ARITHMETIC.  2$ 

two  old  numbers  whose  difference  may  be  made  as  small  as  we 
please.  Numbers  possessing  this  property  are  called  irrational  num- 
bers, in  contradistinction  to  the  numbers  hitherto  defined,  which  are 
termed  rational.  The  considerations  which  before  led  us  to  negative 
rational  numbers,  now  also  lead  us  to  negative  irrational  numbers. 
The  repeated  application  of  addition  and  multiplication  as  of  their 
inverse  processes  to  irrational  numbers,  (numbers  which  though  not 
exactly  equal  to  previously  defined  rational  numbers  may  yet  be 
brought  as  near  to  them  as  we  please,)  again  simply  leads  to  num- 
bers of  the  same  class. 

A  totally  new  domain  of  numbers  is  reached,  however,  when  we 
attempt  to  impart  meaning  to  the  square  roots  of  negative  numbers. 
The  square  root  of  minus  g  is  neither  equal  to  plus  3  nor  to  minus 
3,  since  each  multiplied  by  itself  gives  plus  g,  nor  is  it  equal  to  any 
other  number  hitherto  defined.  Accordingly,  the  square  root  of  minus 
g  is  a  new  number-form,  to  which,  harmoniously  with  the  principle 
of  no  exception,  we  may  give  the  definition  that  (|/ — g)- shall  al- 
ways be  put  equal  to  minus  g.*  Keeping  to  this  definition  we  see 
at  once  that  1/ — a,  where  a  is  any  positive  rational  or  irrational 
number,  is  a  symbol  which  can  be  put  equal  to  the  product  of  V  -|-  a 
by  ]/ — I.  In  extending  to  these  new  numbers  the  rights  of  arith- 
metical citizenship,  in  calling  them  also  "numbers,"  and  so  shaping 
their  definition  that  we  can  reckon  with  them  by  the  same  rules  as 
with  already  defined  numbers,  we  obtain  a  fourth  extension  of  the 
domain  of  numbers  which  has  become  of  the  greatest  importance 
for  the  progress  of  all  branches  of  mathematics.  The  newly  defined 
numbers  are  called  imaginary,  in  contradistinction  to  all  heretofore 
defined,  which  are  called  real.  Since  all  imaginary  numbers  can  be 
represented  as  products  of  real  numbers  with  the  square  root  of 
minus  one,  it  is  convenient  to  introduce  for  this  one  imaginary  num- 
ber some  concise  symbol.  This  symbol  is  the  first  letter  of  the  word 
imaginary,  namely,  /;  so  that  we  can  always  put  for  such  an  ex- 
pression as  1/ —  g,  3 .  /. 

If  we  combine  real  and  imaginary  numbers  by  operations  of  the 

. . ft , 

■)  _        _ 

*  Henceforward  we  shall  use  the  simpler  sign  v'  for  f . 


24  MONISM   IN  ARITHMETIC. 

first  and  second  degree,  always  supposing  that  we  follow  in  our 
reckoning  with  imaginary  numbers  the  same  rules  that  we  do  in 
reckoning  with  real  numbers,  we  always  arrive  again  at  real  or 
imaginary  numbers,  excepting  when  we  join  together  a  real  and  an 
imaginary  number  by  addition  or  its  inverse  operations.  In  this 
case  we  reach  the  symbol  a  -\-  i .  l>,  where  a  and  i?  stand  for  real  num- 
bers. Agreeably  to  the  principle  of  no  exception  we  are  permitted 
to  reckon  with  a  -\-  ib  according  to  the  same  rules  of  computation  as 
with  symbols  previously  defined,  if  for  the  second  power  of  /  we 
always  substitute  minus  i. 

In  the  numerical  conibination  a^  ib,  which  we  also  call  num- 
ber, we  have  fovmd  the  most  general  numerical  form  to  which  the 
laws  of  arithmetic  can  lead,  even  though  we  wished  to  extend  the 
limits  of  arithmetic  still  further.  Of  course,  we  must  represent  to 
ourselves  here  by  a  and  b  either  zero  or  positive  or  negative  rational 
or  irrational  numbers.  If  b  is  zero,  a  +  ib  represents  all  real  num- 
bers ;  if  a  is  zero,  it  stands  for  all  purely  imaginary  numbers.  This 
general  number  a  -f-  ib  is  called  a  complex  tiionber,  so  that  the  com- 
plex number  includes  in  itself  as  special  cases  all  numbers  hereto- 
fore defined.  By  the  introduction  of  irrational,  purely  imaginar)^ 
and  the  still  more  general  complex  numbers,  all  combinations  be- 
come invested  with  meaning  which  the  operations  of  the  third  de- 
gree can  produce.  For  example,  the  fifth  root  of  5  is  an  irrational 
number,  the  logarithm  of  2  to  the  base  10  is  an  irrational  number. 
The  logarithm  of  minus  i  to  the  base  2  is  a  purely  imaginary  num- 
ber; the  fourth  root  of  minus  i  is  a  complex  number.  Indeed,  we 
may  recognise,  proceeding  still  further,  that  every  cojnbifiation  0/ two 
complex  numbers,  by  means  of  any  of  the  operations  of  the  first,  second, 
or  third  degree  will  lead  /«  turn  to  a  complex  tiumber,  that  is  to  say, 
never  furnishes  occasion,  by  application  of  the  principle  of  no  ex- 
ception, for  inventing  new  forms  of  numbers. 

A  certain  limit  is  thus  reached  in  the  construction  of  arithmetic. 
But  such  a  limit  was  also  twice  previousl}'  reached.  After  the  in- 
vestigation of  addition  and  its  inverse  operations,  we  reached  no 
other  numbers  except  zero  and  positive  and  negative  whole  num- 
bers, and  every  combination  of  such   numbers  by  operations  of  the 


MONISM  IN  ARITHMETIC.  25 

first  degree  led  to  no  new  numbers.  After  the  investigation  of  mul- 
tiplication and  its  inverse  operations,  the  positive  or  negative  frac- 
tional numbers  and  "infinitely  great"  were  added,  and  again  we 
could  say  that  the  combination  of  two  already  defined  numbers 
by  operations  of  the  first  and  second  degree  in  turn  also  always 
led  to  numbers  already  defined.  Now  we  have  reached  a  point  at 
which  we  can  say  that  the  combination  of  two  complex  numbers  by 
all  operations  of  the  first,  second,  and  third  degree  must  again 
always  lead  to  complex  numbers  ;  only  that  now  such  a  combina- 
tion does  not  necessarily  always  lead  to  a  single  number,  but  may 
lead  to  many  regularly  arranged  numbers.  For  example,  the  com- 
bination "logarithm  of  minus  one  to  a  positive  base"  furnishes  a 
countless  number  of  results  which  form  an  arithmetical  series  of 
purely  imaginary  numbers.  Still,  in  no  case  now  do  zve  arrive  at  new 
classes  of  numbers.  But  just  as  before  the  ascent  from  multiplication 
to  involution  brought  in  its  train  the  definition  of  new  numbers,  so 
it  is  also  possible  that  some  new  operation  springing  out  of  involution 
as  involution  sprang  from  multiplication  might  furnish  the  germ  of  other 
new  numbers  wiiicJi  are  not  reducible  to  a  -|-  i  h.  As  a  matter  of  fact, 
mathematicians  have  asked  themselves  this  question  and  investi- 
gated the  direct  operation  of  the  fourth  degree,  together  with  its 
inverse  processes.'  The  result  of  their  investigations  was,  that  an 
operation  which  springs  from  involution  as  involution  sprang  from 
multiplication  is  incapable  of  performing  any  real  mathematical  ser- 
vice ;  the  reason  of  which  is,  that  in  involution  the  laws  of  commu- 
tation and  association  do  not  hold.  It  also  further  appeared  that 
the  operations  of  the  fourth  degree  could  not  give  rise  to  new  num- 
bers. No  more  so  can  operations  of  still  higher  degrees.  With 
respect  to  quaternions,  which  many  might  be  disposed  to  regard  as 
new  numbers,  it  will  be  evident  that  though  quaternions  are  valu- 
able means  of  investigation  in  geometry  and  mechanics  they  are  not 
numbers  of  arithmetic,  because  the  rules  of  arithmetic  are  not  un- 
conditionally applicable  to  them. 

The  building  up  of  arithmetic  is  thus  completed.  The  exten- 
sions of  the  domain  of  number  are  ended.  It  only  remains  to  be 
asked  why  the  science  of  arithmetic  appears  in  its  structure  so  logi- 


26  MONISM   IN  ARITHMETIC. 

cal,  natural,  and  unarbitrary ;  why  zero,  negative,  and  fractional 
numbers  appear  as  much  derived  and  as  little  original  as  irrational, 
imaginary,  and  complex  numbers?  We  answer,  wholly  and  alone  in 
virtue  of  the  logical  application  of  the  monistic  principle  of  arith- 
metic, the  principle  of  no  exception. 


ON  THE  NATURE  OF  MATHEMATICAL 
KNOWLEDGE. 

"1\  /f  ATHEMATICALLY  certain  and  unequivocal"  is  a  phrase 
-i-^-^  which  is  often  heard  in  the  sciences  and  in  common  life, 
to  express  the  idea  that  the  seal  of  truth  is  more  deeply  imprinted 
upon  a  proposition  than  is  the  case  with  ordinary  acts  of  knowledge. 
We  propose  to  investigate  in  this  article  the  extent  to  which  math- 
ematical knowledge  really  is  more  certain  and  unequivocal  than 
other  knowledge. 

The  intrinsic  character  of  mathematical  research  and  know- 
ledge is  based  essentially  on  three  properties  :  first,  on  its  conserv- 
ative attitude  towards  the  old  truths  and  discoveries  of  mathematics ; 
secondl}^  on  its  progressive  mode  of  development,  due  to  the  inces- 
sant acquisition  of  new  knowledge  on  the  basis  of  the  old  ;  and  thirdly, 
on  its  self-sufficiency  and  its  consequent  absolute  independence. 

That  mathematics  is  the  most  conservative  of  all  the  sciences 
is  apparent  from  the  incontestability  of  its  propositions.  This  last 
character  bestows  on  mathematics  the  enviable  superiority  that  no 
new  development  can  undo  the  work  of  previous  developments  or 
substitute  new  in  the  place  of  old  results.  The  discoveries  that 
Pythagoras,  Archimedes,  and  Apollonius  made  are  as  valid  to-day 
as  they  were  two  thousand  years  ago.  This  is  a  trait  which  no 
other  science  possesses.  The  notions  of  previous  centuries  regard- 
ing the  nature  of  heat  have  been  disproved.  Goethe's  theory  of 
colors  is  now  antiquated.  The  theory  of  the  binary  combination  of 
salts  was  supplanted  by  the  theory  of  substitution,  and  this,  in  its 
turn,  has  also  given  wa}-  to  newer  conceptions.      Think  of  the  pro- 


28  ON  THE  NATURE  OF  MATHEMATICAL   KNOWLEDGE. 

found  changes  which  the  conceptions  of  theoretical  medicine,  zool- 
ogy, botany,  mineralogy,  and  geology  have  undergone.  It  is  the 
same,  too,  in  the  other  sciences.  In  philology,  comparative  linguis- 
tics, and  history  our  ideas  are  quite  different  from  what  they  formerly 
were. 

In  no  other  science  is  it  so  indispensable  a  condition  that  what- 
ever is  asserted  must  be  true,  as  it  is  in  mathematics.  Whenever, 
therefore,  a  controversy  arises  in  mathematics,  the  issue  is  not 
whether  a  thing  is  true  or  not,  but  whether  the  proof  might  not  be 
conducted  more  simply  in  some  other  way,  or  whether  the  proposi- 
tion demonstrated  is  sufficiently  important  for  the  advancement  of  the 
science  as  to  deserve  especial  enunciation  and  emphasis,  or  finally, 
whether  the  proposition  is  not  a  special  case  of  some  other  and 
more  general  truth  which  is  just  as  easily  discovered. 

Let  me  recall  the  controversy  which  has  been  waged  in  this 
century  regarding  the  eleventh  axiom  of  Euclid,  that  only  one  line 
can  be  drawn  through  a  point  parallel  to  another  straight  line.  This 
discussion  impugned  in  no  wise  the  truth  of  the  proposition  ;  for 
that  things  are  true  in  mathematics  is  so  much  a  matter  of  course 
that  on  this  point  it  is  impossible  for  a  controversy  to  arise.  The 
discussion  merely  touched  the  question  whether  the  axiom  was 
capable  of  demonstration  solely  by  means  of  the  other  propositions, 
or  whether  it  was  not  a  special  property,  apprehensible  only  by 
sense-experience,  of  that  space  of  three  dimensions  in  which  the 
organic  world  has  been  produced  and  which  therefore  is  of  all  others 
alone  within  the  reach  of  our  powers  of  representation.  The  truth 
of  the  last  supposition  affects  in  no  respect  the  correctness  of  the 
axiom  but  simply  assigns  to  it,  in  an  epistemological  regard,  a  dif- 
ferent status  from  what  it  would  have  if  it  were  demonstrable,  as 
was  at  one  time  thought^  without  the  aid  of  the  senses,  and  solely  by 
the  other  propositions  of  mathematics. 

I  may  recall  also  a  second  controversy  which  arose  a  few  de- 
cades ago  as  to  whether  all  continuous  functions  were  differentiable. 
In  the  outcome,  continuous  functions  were  defined  that  possessed 
no  differential  coefficient,  and  it  was  thus  learned  that  certain  truths 
which  were  enunciated   unconditionally  by  Newton,  Leibnitz,  and 


ON  THE  NATURE  OF  MATHEMATICAL  KNOWLEDGE.  2g 

their  mathematical  successors,  required  quahfication.  But  this  did 
not  invalidate  at  all  the  correctness  of  the  method  of  differentiation, 
nor  its  application  in  all  practical  cases;  the  theoretical  specula- 
tions pursued  on  this  subject  simply  clarified  ideas  and  sifted  out 
the  conditions  upon  which  differentiability  depended.  Happily  the 
gifted  minds  who  invent  the  new  methods  and  open  up  the.  new 
paths  of  research  in  mathematics,  are  not  deterred  by  the  fear  that 
a  subsequent  generation  gifted  with  unusual  acumen  will  spy  out 
isolated  cases  in  which  their  methods  fail.  Happily  the  creators 
of  the  differential  calculus  pushed  onward  without  a  thought  that  a 
critical  posterity  would  discover  exceptions  to  their  results.  In 
every  great  advance  that  mathematics  makes,  the  clarification  and 
scrutinisation  of  the  results  reached  are  reserved  necessarily  for  a 
subsequent  period,  but  with  it  the  demonstration  of  those  results  is 
more  rigorously  established.  Despite  all  this,  however,  in  no  sci- 
ence does  cognition  bear  so  unmistakably  the  imprint  of  truth  as  in 
pure  mathematics.  And  this  fact  bestows  on  mathematics  its  con- 
servative character. 

This  conservative  character  again  is  displayed  in  the  objects  of 
mathematical  research.  The  physician,  the  historian,  the  geogra- 
pher, and  the  philologist  have  to-day  quite  different  fields  of  inves- 
tigation from  what  they  had  centuries  ago.  In  mathematics,  too, 
every  new  age  gives  birth  to  new  problems,  arising  partly  from  the 
advance  of  the  science  itself,  and  partly  also  from  the.  advance  of 
civilisation,  where  improvements  in  the  other  sciences  bring  in  their 
train  new  problems  that  are  constantly  taxing  afresh  the  resources 
of  mathematics.  But  despite  all  this,  in  mathematics  more  than  in 
any  other  science  problems  exist  that  have  played  a  role  for  hun- 
dreds, nay,  for  thousands  of  years. 

In  the  oldest  mathematical  manuscript  which  we  possess,  the 
Rhind  Papyrus  of  the  British  Museum,  which  dates  back  to  the 
eighteenth  century  before  Christ,  and  whose  decipherment  we  owe 
to  the  industry  of  Eisenlohr,  we  find  an  attempt  to  solve  the  prob- 
lem of  converting  a  circle  into  a  square  of  equal  area,  a  problem 
whose  history  covers  a  period  of  three  and  a  half  thousand  years. 
For  it  was  not  until  1882  that  a  rigorous  proof  was  given  of  the  im- 


30         ON  -JHE  NATURE  OF  MATHEMATICAL  KNOWLEDGE. 

possibility  of  solving  this  problem  exactly  by  the  use  of  straight 
edge  and  compasses  alone.      (See  pp,  ii6,  141-143.) 

It  is,  of  course,  the  insoluble  problems  that  have  the  longest 
history;  partly  because  it  is  harder  to  show  that  a  thing  is  impos- 
sible than  that  it  is  possible,  and,  on  the  other  hand,  because  prob- 
lems that  have  long  defied  solution  are  ever  evoking  anew  the  spirit 
of  inquiry  and  the  ambition  of  mathematicians,  and  because  the  un- 
certainty of  insolubility  lends  to  such  problems  a  peculiar  charm. 
Of  the  geometrical  problems  that  have  occupied  competent  and  in- 
competent minds  from  the  time  of  the  ancient  Greeks  to  the  present 
may  be  mentioned  in  addition  to  the  squaring  of  the  circle  two 
others  that  are  also  perhaps  well-known  to  educated  readers,  at  least 
by  name  :  the  trisection  of  the  angle  and  the  Delic  problem  of  the 
duplication  of  the  cube.  All  three  problems  involve  the  condition, 
which  is  often  overlooked  by  lay  readers,  that  only  straight  edge 
and  compasses  shall  be  employed  in  the  constructions.  In  the  tri- 
section of  the  angle  any  angle  is  assigned,  and  it  is  required  to  find 
the  two  straight  lines  which  divide  the  angle  into  three  equal  parts. 
In  the  Delic  problem  the  edge  of  a  cube  is  given  and  the  edge  of  a 
second  cube  is  sought,  containing  twice  the  volume  of  the  first  cube. 
In  Greece,  in  the  golden  age  of  the  sciences,  when  all  scholars  had 
to  understand  mathematics,  it  was  a  fashionable  requisite  almost  to 
have  employed  oneself  on  these  famous  problems. 

Fortunately  for  us,  these  problems  were  insoluble.  For  in  their 
ambition  to  conquer  them  it  came  to  pass  that  men  busied  themselves 
more  and  more  with  geometry,  and  in  this  way  kept  constantly  dis- 
covering new  truths  and  developing  new  theories,  all  of  which  per- 
haps might  never  have  been  done  if  the  problems  had  been  soluble 
and  had  early  received  their  solutions.  Thus  is  the  struggle  after 
truth  often  more  fruitful  than  the  actual  discovery  of  truth.  So,  too, 
although  in  a  slightly  different  sense,  the  apophthegm  of  Lessing  is 
confirmed  here,  that  the  search  for  truth  is  to  bo  preferred  to  its 
possession. 

Whilst  the  three  above-named  problems  are  now  acknowledged 
to  be  insoluble,  and  have  ceased,  therefore,  to  stimulate  mathemat- 
ical inquiry,   there  are  of  course   other  problems  in   mathematics 


ON  THE  NATURE  OF  MATHEMATICAL  KNOWLEDGE.  3  I 

whose  solution  has  been  sought  for  a  long  time,  but  not  yet  reached, 
and  in  the  case  of  which  there  is  no  reason  for  supposing  that  they 
are  insoluble.  Of  such  problems  the  two  following  perhaps  have 
found  their  way  out  of  the  isolated  circles  of  mathematicians  and 
have  become  more  or  less  known  to  other  scholars.  I  refer  to  the 
astronomical  Problem  of  Three  Bodies  and  to  the  problem  of  the 
frequency  of  prime  numbers.  The  first  of  these  two  problems  as- 
sumes three  or  more  heavenly  bodies  whose  movements  are  mutually 
influenced  by  one  another  according  to  Newton's  law  of  gravitation, 
and  requires  the  exact  determination  of  the  path  which  each  body 
describes.  The  second  problem  requires  the  construction  of  a  for- 
mula which  shall  tell  how  many  prime  numbers  there  are  below  a 
certain  given  number.  So  far  these  two  problems  have  been  solved 
only  approximately,  and  not  with  absolute  mathematical  exactness. 
If  the  eternal  and  inviolable  correctness  of  its  truths  lends  to 
mathematical  research,  and  therefore  also  to  mathematical  knowl- 
edge, a  conservative  character,  on  the  other  hand,  by  the  continuous 
outgrowth  of  new  truths  and  methods  from  the  old,  progressiveness 
is  also  one  of  its  characteristics.  In  marvellous  prcffusion  old  knowl- 
edge is  augmented  by  new,  which  has  the  old  as  its  necessary  con- 
dition, and,  therefore,  could  not  have  arisen  had  not  the  old  pre- 
ceded it.  The  indestructibility  of  the  edifice  of  mathematics  renders 
it  possible  that  the  work  can  be  carried  to  ever  loftier  and  loftier 
heights  without  fear  that  the  highest  stories  shall  be  less  solid  and 
safe  than  the  foundations,  which  are  the  axioms,  or  the  lower  sto- 
ries, which  are  the  elementary  propositions.  But  it  is  necessary  for 
this  that  all  the  stones  should  be  properly  fitted  together ;  and  it 
would  be  idle  labor  to  attempt  to  lay  a  stone  that  belonged  above  in 
a  place  below.  A  good  example  of  a  stone  of  this  character  belong- 
'•ingin  what  is  now  the  uppermost  laj'er  of  the  edifice,  is  Linde- 
mann's  demonstration  of  the  insolubility  of  the  quadrature  of  the 
circle,  a  demonstration  of  which  interesting  simplifications  have 
been  given  by  several  mathematicians,  including  Weierstrass  and 
Felix  Klein.  Lindemann's  demonstration  could  not  have  been  pro- 
duced in  the  preceding  century,  because  it  rests  necessarily  on  theo- 
ries whose  development  falls  in   the  present  century.      It  is  true, 


32  ON  THE  NATURE  OF  MATHEMATICAL  KNOWLEDGE. 

Lambert  succeeded  in  1761  in  demonstrating  the  irrationality  of  the 
ratio  of  the  circumference  of  a  circle  to  its  diameter,  or,  which  is 
the  same  thing,  the  irrationality  of  the  ratio  of  the  area  of  a  circle 
to  the  area  of  the  square  on  its  radius.  Afterwards,  Lambert  also 
supplied  a  proof  that  it  was  impossible  for  this  ratio  to  be  the  square 
root  of  a  rational  number.  But  this  was  the  first  step  only  in  a  long 
journey.  The  attempt  to  prove  that  the  old  problem  is  insoluble 
was  still  destined  to  fail.  An  astounding  mass  of  mathematical  in- 
vestigations were  necessary  before  the  demonstration  could  be  suc- 
cessfull3^  accomplished. 

As  we  see,  the  majority  of  the  mathematical  truths  now  pos- 
sessed by  us  presuppose  the  intellectual  toil  of  many  centuries.  A 
mathematician,  therefore,  who  wishes  to-day  to  acquire  a  thorough 
understanding  of  modern  research  in  this  department,  must  think 
over  again  in  quickened  tempo  the  mathematical  labors  of  several 
centuries.  This  constant  dependence  of  new  results  on  old  ones 
stamps  mathematics  as  a  science  of  uncommon  exclusiveness  and 
renders  it  generally  impossible  to  lay  open  to  uninitiated  readers  a 
speedy  path  to  the  apprehension  of  the  higher  mathematical  truths. 
For  this  reason,  too,  the  theories  and  results  of  mathematics  are 
rarely  adapted  for  popular  presentation.  There  is  no  royal  road  to 
the  knowledge  of  mathematics,  as  Euclid  once  said  to  the  first 
Egyptian  Ptolemy.  This  same  inaccessibility  of  mathematics,  al- 
though it  secures  for  it  a  lofty  and  aristocratic  place  among  the  sci- 
ences, also  renders  it  odious  to  those  who  have  never  learned  it,  and 
who  dread  the  great  labor  involved  in  acquiring  an  understanding 
of  the  questions  of  modern  mathematics.  Neither  in  the  languages 
nor  in  the  natural  sciences  are  the  investigations  and  results  so 
closely  interdependent  as  to  make  it  impossible  to  acquaint  the  un- 
initiated student  with  single  branches  or  with  particular  results  of 
these  sciences,  without  causing  him  to  go  through  a  long  course  of 
preliminary  study. 

The  third  trait  which  distinguishes  mathematical  research  is  its 
self-sufficiency.  In  philology  the  field  of  inquiry  is  the  organic  one 
of  languages,  and  philology,  therefore,  is  dependent  in  its  investiga- 
tions on  the  mode  of  development  of  languages,  which  is  more  or 


ON  THE   NATURE  OF  MATHEMATICAL  KNOWLEDGE.  33 

less  accidental.  Its  task  is  connected  with  something  which  is  given 
to  it  from  without  and  which  it  cannot  alter.  It  is  much  the  same 
with  the  science  of  history,  which  must  contemplate  the  history  of 
mankind  as  it  has  actually  occurred.  Also  zoology,  botany,  mineral- 
ogy, geology,  and  chemistry  work  with  given  data.  In  order  not 
to  become  involved  in  futile  speculations  the  last-mentioned  sci- 
ences are  constantly  and  inevitably  obliged  to  revest  to  observa- 
tions by  the  senses.  It  is  then  their  task  to  link  together  these  in- 
dividual observations  by  bonds  of  causality  and  in  this  way  to 
erect  from  the  single  stones  an  edifice,  the  view  of  which  will  render 
it  easier  for  limited  human  intelligence  to  comprehend  nature. 
Physics  of  all  sciences  stands  nearest  to  mathematics  in  this  respect, 
because  unlike  the  other  sciences  she  is  generally  in  need  of  only  a 
few  observations  in  order  to  proceed  deductively.  But  physics, 
too,  must  resort  to  observations  of  nature,  and  could  not  do  without 
them  for  any  length  of  time. 

Mathematics  alone,  after  certain  premises  have  been  perma- 
nently established,  is  able  to  pursue  its  course  of  development  in- 
dependently and  unmindful  of  things  outside  of  it.  It  can  leave 
entirely  unnoticed,  questions  and  influences  emanating  from  the 
outer  world,  and  continue  nevertheless  its  development. 

As  regards  geometry,  the  first  beginnings  of  this  science  did 
indeed  take  their  origin  in  the  requirements  of  practical  life.  But 
it  was  not  long  before  it  freed  itself  from  the  restrictions  of  the  prac- 
tical art  to  which  it  owed  its  birth.  Herodotus  recounts  that  geom- 
etry had  its  origin  in  Egypt  where  the  inundations  of  the  Nile  ob- 
literated the  boundaries  of  the  riparian  estates,  and  by  giving  rise 
to  frequent  disputes  constantly  compelled  the  inhabitants  to  compare 
the  areas  of  fields  of  different  shapes.  But  with  the  early  Greek 
mathematicians,  who  were  the  heirs  of  the  Egyptian  art  of  measure- 
ment, geometry  appeared  as  a  science  which  men  pursued  for  its 
own  sake  without  a  thought  of  how  their  intellectual  discoveries 
could  be  turned  to  practical  account. 

Nevertheless,  although  the  workers  in  the  domain  of  pure  math- 
ematics are  not  stimulated  by  the  thought  that  their  researches  are 
likely  to  be  of  practical  value,  yet  that  result  is  still  frequently  real- 


34  ON  THE  NATURE  OF  MATHEMATICAL  KNOWLEDGE. 

ised,  often  after  the  lapse  of  centuries.  The  history  of  mathe- 
matics shows  numerous  instances  of  mathematical  results  which 
were  originally  the  outcome  of  a  mere  desire  to  extend  the  science, 
suddenly  receiving  in  astronomy,  mechanics,  or  in  physics  practical 
applications  which  their  originators  could  scarce  have  dreamt  of. 
Thus  Apollonius  erected  in  ancient  times  the  stately  edifice  of  the 
properties  of  conic  sections,  without  having  any  idea  that  the  plan- 
ets moved  about  the  sun  in  conic  sections,  and  that  a  Kepler  and  a 
Newton  were  one  day  to  come  who  should  apply  these  properties  to 
explaining  and  calculating  the  motions  of  the  planets  about  the 
sun.  The  question  of  the  practical  availability  of  its  results  in  other 
fields  has  at  no  period  exercised  more  than  a  subordinate  influence 
on  mathematical  inquiry.  Particularly  is  this  true  of  inodfrn  math- 
ematical research,  whether  the  same  consist  in  the  extended  de- 
velopment of  isolated  theories  or  in  uniting  under  a  higher  point  of 
.view  theories  heretofore  regarded  as  different.* 

This  independence  of  its  character  has  rendered  the  results  of 
pure  mathematics  independent  also  of  the  accidental  direction  which 
the  development  of  civilisation  has  taken  on  our  planet  ;  so  that 
the  remark  is  not  altogether  without  justification,  that  if  beings  en- 
dowed with  intelligence  existed  on  other  planets,  the  truths  of  math- 
ematics would  afford  the  only  basis  of  an  understanding  with  them. 
Uninterruptedly  and  wholly  from  its  own  resources  mathematics  has 
built  itself  up.  It  is  scarcely  credible  to  a  person  not  versed  in  the 
science,  that  mathematicians  can  derive  satisfaction  from  the  com- 
fortless and  wearisome  operation  of  heaping  up  demonstration  on 
demonstration,  of  rivetting  truth  on  truth,  and  of  tormenting  them- 
selves with  self-imposed  problems,  whose  solution  stands  no  one  in 
stead,  and  affords  satisfaction  to  no  one  but  the  solver  himself.  Yet 
this  self-sufficiency  of  mathematicians  becomes  a  little  more  intel- 
ligible when  we  reflect  that  the  progress  which  has  been  made,  par- 
ticularly in  the  last  few  decades,  and  which  is  uninfluenced  from 
without,  does  not  consist  solely  in  the  accumulation  of  new  truths 


*Cf.  Felix  Klein,   "Remarks  Given  at  the  Opening  of  the  Mathematical  and 
Astronomical  Congress  at  Chicago. "      The  Moiiist  (Vol.  IV,  No.  i,  October,  1893). 


ON  THE  NATURE  OF  MATHEMATICAL  KNOWLEDGE.  35 

and  in  the  enunciation  of  new  problems,  nor  solely  in  deductions 
and  solutions,  but  culminates  rather  in  the  discovery  of  new  meth- 
ods and  points  of  view  in  which  the  old  disconnected  and  isolated 
results  appear  suddenly  in  a  new  connexion  or  as  different  interpre- 
tations of  a  common  fundamental  truth,  or  finally,  as  a  single  or- 
ganic whole. 

Thus,  for  example,  the  idea  of  representing  imaginary  and  com- 
plex numbers  in  a  plane,  two  apparently  different  branches,  the  theory 
of  dividing  the  circumference  of  a  circle  into  any  given  number  of 
equal  parts,  and  the  theory  of  the  solutions  of  the  equation  x"=zi, 
have  been  made  to  exhibit  an  extremely  simple  connexion  with  one 
another  which  enables  us  to  express  many  a  truth  of  algebra  in  a 
corresponding  truth  of  geometry  and  vne  versa.  Another  example  is 
afforded  by  the  discovery  which  we  chiefly  owe  to  Alfred  Clebsch, 
of  the  relation  which  subsists  between  the  higher  theory  of  func- 
tions and  the  theory  of  algebraic  curves,  a  relation  which  led  to  the 
discovery  of  the  condition  under  which  two  curves  can  be  co-ordi- 
nated  to  each  other,  point  for  point,  and  hence  also  adequately  rep- 
resented on  each  other.  Of  course  such  combinations  and  exten- 
sions of  view  possess  a  much  greater  charm  for  the  mathematician 
than  the  mere  accumulation  of  truths  and  solutions,  whose  fascina- 
tion consists  entirely  in  their  truth  or  correctness. 

From  these  three  cardinal  characteristics,  now,  which  distin- 
guish mathematical  research  from  research  in  other  fields,  we  may 
gather  at  once  the  three  positive  characteristics  that  distinguish 
mathematical  knowledge  from  other  knowledge.  They  may  be  briefly 
expressed  as  follows ;  first,  mathematical  knowledge  bears  more 
distinctly  the  imprint  of  truth  on  all  its  results  than  any  other 
kind  of  knowledge  ;  secondly,  it  is  always  a  sure  preliminary  step  to 
the  attainment  of  other  correct  knowledge  ;  thirdly,  it  has  no  need 
of  other  knowledge.  Naturally,  however,  there  are  associated  with 
these  characteristics  which  place  mathematical  knowledge  high 
above  all  other  knowledge,  other  characteristics  which  somewhat 
counterbalance  the  great  superiority  which  mathematics  thus  ap- 
pears to  have  over  the  other  sciences.  In  order  to  show  more  dis- 
tinctly the  nature  of  these  characteristics,  which  we  prefer  to  call 


36  ON  THE  NATURE  OF  MATHEMATICAL  KNOWLEDGE. 

negative,  we  shall  select  and  confine  our  remarks  to  a  branch  which 
is  commonly  taken  to  be  synon3'mous  with  mathematics,  narrjely,  to 
arithmetic  in  the  broadest  sense  of  the  word. 

The  subject  of  inquiry  in  arithmetic  is  numbers  and  their  com- 
binations. On  this  account  arithmetic  is,  of  all  sciences,  most  free 
from  what  lies  outside  its  boundaries.  Perception  by  the  senses  is 
necessary  only  in  an  extremely  insignificant  measure  for  the  under- 
standing of  its  definitions  and  premises.  It  is  possible  to  acquaint 
a  person  who  lacks  both  sight  and  hearing  with  the  fundamental 
principles  of  arithmetic  solely  by  the  medium  of  "time."  Such  a 
person  needs  only  the  sense  of  feeling.  By  slight  excitations  of  his 
skin,  induced  at  equal  or  unequal  intervals  of  time,  he  can  be  led  to 
the  notion  of  differences  of  time  and  hence  also  to  the  notion  of  dif- 
ferences of  number.  Uninfluenced  by  matter  and  force,  independ- 
ently, too,  of  the  properties  of  geometrical  magnitudes,  arithmetic 
could  be  conducted  solely  by  its  own  intrinsic  potencies  to  its  high- 
est goals,  drawing  deductively  truth  from  truth,  without  a  break. 

But  what  sort  of  a  science  should  we  arrive  at  by  this  method 
of  procedure?  Nothing  but  a  gigantic  web  of  self-evident  truths. 
For,  once  we  admit  the  first  notions  and  premises  to  which  a  man 
thus  bereft  of  his  senses  can  be  led,  we  are  compelled  of  necessity 
also  to  admit  the  derivative  results  of  arithmetic.  If  the  beginnings 
of  arithmetic  appear  self-evident,  the  rest  of  it,  too,  bears  this 
character.  Owing  to  this  deductive  character  of  arithmetic,  and  to 
its  exemption  from  influence  from  without,  this  science  appears  to 
one  person  extremely  attractive,  while  to  another  it  appears  ex- 
tremely repulsive,  according  as  each  is  constituted.  Be  that  as  it 
may,  however,  a  finished  and  complete  science  of  this  character 
subserves  no  purpose  in  the  comprehension  of  the  world,  or  in  the 
advancement  of  civilisation.  Hence,  an  arithmetic  which  heaps  up 
theorem  on  theorem  with  never  a  thought  of  how  its  results  are  to 
be  turned  to  practical  account  in  the  acquisitioij  of  knowledge  in 
other  fields,  resembles  an  inquisitive  physician,  who,  taking  up  his 
abode  in  a  desert,  should  arrive  there  at  momentous  results  in 
bacteriology,  but  should  bear  them  with  him  to  his  grave,  without 
their  ever  redounding  to   the  benefit  of  humanity.      The  value  of 


ON  THE  NATURE  OF  MATHEMATICAL  KNOWLEDGE.  37 

arithmetical  knowledge  lies  entirely  in  its  applications.  But  this 
constitutes  no  reason  why  many  mathematicians,  pursuing  their 
purely  deductive  bent  of  mind,  should  not  devote  themselves  ex- 
clusively to  pure  arithmetical  developments  and  leave  it  to  others 
at  the  proper  time  to  turn  to  the  material  profit  of  the  world  the 
capital  which  they  have  garnered. 

Geometry,  on  the  other  hand,  must  have  recourse  in  a  much 
higher  degree  than  arithmetic  to  the  outside  world  for  its  first  notions 
and  premises.  The  axioms  of  geometry  are  nothing  but  facts  of  ex- 
perience perceived  by  our  senses.  The  geometry  which  Bolyai,  Lo- 
bach^vski.  Gauss,  Riemann,  and  Helmholtz  created  and  which  is 
both  independent  of  the  eleventh  axiom  of  Euclid  and  perfectly 
free  from  self-contradictions,  has  supplied  an  epistemological  dem- 
onstration that  geometry  is  a  science  that  rests  on  the  observation 
of  nature,  and  therefore  in  the  correct  sense  of  the  word,  is  a  natu- 
ral science. 

.  Yet  what  a  difference  there  is,  for  instance,  between  geometry 
and  chemistry!  Both  derive  their  constructive  materials  from  sense- 
perception.  But  whilst  geometry  is  compelled  to  draw  only  its  first 
results  from  observation  and  is  then  in  a  position  to  move  forward 
deductively  to  other  results  without  being  under  the  necessity  of 
making  fresh  observations,  chemistry  on  the  other  hand  is  still 
compelled  to  make  observations  and  to  have  recourse  to  nature. 

It  follows,  therefore,  that  a  given  act  of  geometrical  knowledge 
and  a  given  act  of  chemical  knowledge  are  with  respect  to  the  cer- 
tainty of  the  truth  they  contain  not  qualitatively  but  only  quantita- 
tively different.  In  chemistry  the  probability  of  error  is  greater 
than  in  geometry,  because  more  numerous  and  more  difficult  ob- 
servations have  to  be  made  there  than  in  geometry,  where  only  the 
very  first  premises,  which  no  man  with  sound  senses  could  ever  im- 
pugn, rest  on  observation. 

The  preceding  reflexions  deprive  mathematical  knowledge  of 
that  degree  of  certainty  and  incontestability  which  is  commonly 
attributed  to  it  when  we  say  a  thing  is  '' mathematically  certain." 
This  certainty  is  lessened  still  more  as  we  pass  to  the  semi-math- 
ematical sciences,  where  mechanics  has  the  first  claim  to  our  at- 


38  ON  THE  NATURE  OF  MATHEMATICAL  KNOWLEDGE. 

tention.  All  the  notions  of  mechanics,  and  consequently  of  all 
the  other  departments  of  physics,  are  composed,  by  multiplication 
or  division,  of  three  fundamental  notions — length,  time,  and  mass. 
That  is  to  say,  to  the  notions  of  geometry  resting  on  length  and  its 
powers,  two  other  fundamental  notions,  time  and  mass,  are  added, 
which,  joined  to  that  of  length,  lead  to  the  notions  of  force,  work, 
horse-power,  atmospheric  pressure,  etc.  The  knowledge  of  me- 
chanics, thus,  highly  certain  though  it  be,  is  rendered  less  certain 
than  that  of  geometry  and  a  fortiori  than  that  of  arithmetic.  The 
uncertainty  of  knowledge  continues  to  increase  in  branches  which 
^re  still  more  remote  from  mathematics,  owing  to  the  increasing  com- 
plexity of  the  observational  material  which  must  here  be  put  to  the 
test. 

Still,  although  mathematical  knowledge  does  not  lead  to  abso- 
lutely certain  results,  it  yet  invests  known  results  with  incomparably 
greater  trustworthiness  than  does  the  knowledge  of  the  other  sci- 
ences. But  after  all,  it  remains  a  useless  accumulation  of  capital 
so  long  as  it  is  not  turned  to  practical  account  in  other  sciences, 
such  as  metaphysics,  physics,  chemistry,  biology,  political  economy, 
etc.  Hence  also  arises  an  obligation  on  the  part  of  the  other  sci- 
ences, so  to  shape  their  problems  and  investigations  that  they  can 
be  made  susceptible  of  mathematical  treatment.  Then  will  mathe- 
matics gladly  perform  her  duty.  The  moment  a  science  has  ad- 
vanced far  enough  to  permit  of  the  mathematical  formulation  of  its 
problems,  mathematics  will  not  be  slow  to  treat  and  to  solve  these 
problems.  Mathematical  knowledge,  aristocratic  as  it  may  appear 
by  the  greater  certainty  of  its  results,  will,  so  far  as  the  advance- 
ment of  human  kind  is  concerned,  never  be  more  than  a  useless 
mass  of  self-evident  truths,  unless  it  constantly  places  itself  in  the 
service  of  the  other  sciences. 


THE  MAGIC  SQUARE. 

I. 

INTRODUCTORY. 

AMONG  the  philosophies  of  modern  times  there  is  none  which 
■^-^  has  emphasised  so  much  the  importance  of  form  and  formal 
thought  as  the  monism  of  T/ie  Monist.  An  expression  of  this  phi- 
losophy is  found  in  the  following  passages  : 

"  The  order  that  prevails  among  the  facts  of  reality  is  due  to  the  laws  of  form. 
Upon  the  order  of  the  world  depends  its  cognisability. 

"  .  .  .  .  The  laws  of  form  are  no  less  eternal  than  are  matter  and  energy  and 
'  Verily  I  say  unto  you,  till  heaven  and  earth  pass,  one  jot  or  one  tittle  shall  in  no 
wise  pass  from  the  law  ! ' 

"The  laws  of  form  and  their  origin  have  been  a  puzzle  to  all  philosophers. 
'  Ay,  there's  the  rub  !  '  The  difficulties  of  Hume's  problem  of  causation,  of  Kant's 
a  priori,  of  Plato's  ideas,  of  Mill's  method  of  deduction,  etc.,  etc.,  all  arise  from  a 
one-sided  view  of  form  and  the  laws  of  form  and  formal  thought." 

Considering  the  great  results  which  engineering  and  other  ap- 
plied sciences  accomplish  through  the  assistance  of  mathematics, 
we  must  confess  that  the  forms  of  thought  are  wonderful  indeed, 
and  it  is  not  at  all  astonishing  that  the  primitive  thinkers  of  man- 
kind when  the  importance  of  the  laws  of  formal  thought  in  some 
way  or  another  first  dawned  on  their  minds,  attributed  magic  powers 
to  numbers  and  geometrical  figures. 

We  shall  devote  the  following  pages  to  a  brief  review  of 
magic  squares,  the  consideration  of  which  has  made  many  a  man 
believe  in  mysticism.  And  yet  there  is  no  mysticism  about  them 
unless  we  either  consider  everything  mystical,  even  that  twice  two 
is  four,  or  join  the  sceptic  in  his  exclamation  that  we  can  truly  not 


ALBERT  DURERS  ENGRAVING 


MELANCHOLY 


Genius  of  the  Industrial  Science  of  Mechanics. 


THE  MAGIC  SQUARE.  41 

know  whether  twice  two  might  not  be  five  in  other  spheres  of  the 
universe. 

The  author  of  the  short  article  on  ' '  Magic  Squares  "  in  the  Eng- 
lish Cyclopaedia  (Vol.  Ill,  p.  415),  presumably  Prof.  De  Morgan, 
says  : 

"  Though  the  question  of  magic  squares  be  in  itself  of  no  use,  yet  it  belongs  to 
a  class  of  problems  which  call  into  action  a  beneficial  species  of  investigation.  With- 
out  laying  down  any  rules  for  their  construction,  we  shall  content  ourselves  with 
destroying  their  magic  quality,  and  showing  that  the  non-existence  of  such  squares 
would  be  much  more  surprising  than  their  existence." 

This  is  the  point.  There  obtains  a  symphonic  harmony  in 
mathematics  which  is  the  more  startling  the  more  obvious  and  self- 
evident  it  appears  to  him  who  understands  the  laws  that  produce  this 
symphonic  harmony. 

*  * 

On  the  wood-cut  named  "Melancholia"*  of  the  famous  Nurem- 
berg painter,  Albrecht  Diirer,  is  found  among  a  number  of  other 


*  The  term  melancholy  meant  in  DUrer's  time,  as  it  did  also  in  Shakespeare's 
and  Milton's,  "  thought  or  thoughtfulness."     Says  Milton  in  //  Penseroso  : 

"  Hail,  thou  Goddess,  sage  and  holy. 
Hail  divinest  melancholy 
Whose  saintly  visage  is  too  bright 
To  hit  the  sense  of  human  sight, 
And  therefore  to  our  weaker  view 
O'erlaid  with  black,  staid  Wisdom's  hue. — I,  la. 

Thought  that  does  not  lead  to  action  produces  a  gloomy  state  of  mind.  Thought- 
fulness  which  cannot  find  a  way  out  of  itself  is  that  melancholy  which  engenders 
weakness, — a  truth  which  is  illustrated  in  Hamlet.  Shakespeare  still  uses  the  words 
thought  and  melancholy  as  synonyms,  saying  : 

"  The  native  hue  of  resolution 
Is  sicklied  o'er  with  the  pale  cast  of  thought." 

Durer's  melancholy  does  not  represent  the  gloominess  of  thought,  but  the  power 
of  invention.  Soberness  and  even  a  certain  sadness  are  considered  only  as  an  ele- 
ment of  this  melancholy,  but  on  the  whole  the  genius  of  thought  appears  bright, 
self-possessed,  and  strong. 

Durer  represents  the  Science  of  Mechanical  Invention  as  a  winged  female  figure 
musing  over  some  problem.  Scattered  on  the  floor  around  her  lie  some  of  the  sim- 
ple tools  used  in  the  sixteenth  century.  A  ladder  leans  against  the  house,  that  as- 
sists in  climbing  otherwise  inaccessible  heights.  A  scale,  an  hour-glass,  a  bell,  and 
the  magic  square  are  hanging  on  the  wall  behind  her. 

At  a  distance  a  bat-like  creature,  being  the  gloom  of  melancholy,  hovers  in  the 
air  like  a  dark  cloud,  but  the  sun  rises  above  the  horizon,  and  at  the  happy  middle 
between  these  two  extremes  stands  the  rainbow  of  serene  hope  and  cheerful  confi- 
dence. 


42 


THE  MAGIC  SQUARE. 


emblems,  which  the  reader  will  notice  in  our  reproduction  of  the  cut, 
the  subjoined  square.   This  arrangement  of  the  sixteen  natural  num- 


I 

14 

15 

^ 

12 

7 

6 

9 

8 

II 

10 

5 

13 

2 

3 

16 

Fig.  I.     * 

bers  from  i  to  16  possesses  the  remarkable  property  that  the  same 
sum  34  will  always  be  obtained  whether  we  add  together  the  four 
figures  of  any  of  the  horizontal  rows  or  the  four  of  any  vertical  row 
or  the  four  which  lie  in  either  of  the  two  diagonals.  Such  an  ar- 
rangement of  numbers  is  termed  a  magic  square,  and  the  square 
which  we  have  reproduced  above  is  the  first  magic  square  which  is 
met  with  ifi  the  Christian  Occident. 

Like  chess  and  many  of  the  problems  founded  on  the  figure  of 
the  chess-board,  the  problem  of  constructing  a  magic  square  also 
probably  traces  its  origin  to  Indian  soil.  From  there  the  problem 
found  its  way  among  the  Arabs,  and  by  them  it  was  brought  to  the 
Roman  Orient.  Finally,  since  Albrecht  Diirer's  time,  the  scholars  of 
Westwn  Europe  also  have  occupied  themselves  with  methods  for 
the  construction  of  squares  of  this  character. 

The  oldest  and  the  simplest  magic  square  consists  of  the  quad- 
ratic arrangement  of  the  nine  numbers  from  i  to  9  in  such  a  man- 
ner that  the  sum  of  each  horizontal,  vertical,  or  diagonal  row,  al- 
ways remains  the  same,   namely  15.      This  square  is  the  adjoined. 


276 
9     5      I 

4      3      8 


Fig.  2. 

Here,  we  will  find,  15  always  comes  out  whether  we  add  2  and  7  and 
6,  or  9  and  5  and  i,  or  4  and  3  and  8,  or  2  and  9  and  4,  or  7  and  5 
and  .3,  or  6  and  i  and  8,  or  2  and  5  and  8,  or  6  and  5  and  4. 

The  question  naturally  presents  itself,  whether  this  condition 
of  the  constant  equality  of  the  added  sum  also  remains  fulfilled  when 
the  numbers  are  assigned  different  places.      It  may  be  easily  shown 


THE  MAGIC  SQUARE. 


43 


however  that  5  necessarily  must  occupy  the  middle  place,  and  that 
the  even  niimbers  must  stand  in  the  corners.  This  being  so,  there 
are  but  7  additional  arrangements  possible,  which  differ  from  the 
arrangement  above  given  and  from  one  another  only  in  the  respect 
that  the  rows  at  the  top,  at  the  left,  at  the  bottom,  and  at  the  right, 
exchange  places  with  one  another  and  that  in  addition  a  mirror  be 
imagined  present  with  each  arrangement.  So  too  from  Diirer's 
square  of  4  times  4  places,  by  transpositions,  a  whole  set  of  new 
correct  squares  may  be  formed.  A  magic  square  of  the  4  times  4 
numbers  from  i  to  16  is  formed  in  the  simplest  manner  as  follows. 
We  inscribe  the  numbers  from  i  to  16  in  their  natural  order  in  the 
squares,  thus  : 


I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

12 

13 

14 

15 

16 

Fig.  3. 

We  then  leave  the  numbers  in  the  four  corner-squares,  viz.  i,  4,  13',, 
16,  as  well  also  as  the  numbers  in  the  four  middle-squares,  viz.  6, 
7,  10,  II,  in  their  original  places  ;  and  in  the  place  of  the  remaining, 
eight  numbers,  we  write  the  complements  of  the  same  with  respect 
to  17:  thus  15  instead  of  2,  14  instead  of  3,  12  instead  of  5,  9  in- 
stead of  8,  8  instead  of  9,  5  instead  of  12,  3  instead  of  14,  and  2  in- 
stead of  15.      We  obtain  thus  the  magic  square 


I 

15 

14 

4 

=34 

12 

6 

7 

9 

=34' 

8 

10 

II 

5 

=34 

13 

3 

2 

16 

=34 

34 

34      34      34 

Fit 

;•  4- 

from  which  the  same  sum  34  always  results.  It  is  an  interesting 
property  of  this  square  that  any  four  numbers  which  form  a  rectangle 
or  square  about  the  centre  also  always  give  the  same  sum  34  ;  for 
example,    i,  4,  13,  16,  or  6,  7,  10,  11,  or  15,  14,  3,  2,  or  12,  9,  5,  8,, 


44 


THE  MAGIC  SQUARE. 


or  15,  8,  2,  9,  or  14,  12,  3,  5.  We  may  easily  convince  ourselves 
that  this  square  is  obtainable  from  the  square  of  Diirer  by  inter- 
changing with  one  another  the  two  middle  vertical  rows. 


EARLY  METHODS    FOR   THE   CONSTRUCTION    OF  ODD-NUMBERED 

SQUARES. 

Since  early  times  rules  have  also  been  known  for  the  construc- 
tion of  magic  squares  of  more  than  3  times  3,  or  4  times  4  spaces. 
In  the  first  place,  it  is  easy  to  calculate  the  sum  which  in  the  case 
of  any  given  number  of  cells  must  result  from  the  addition  of  each 
row.  We  take  the  determinate  number  of  cells  in  each  side  of  the 
square  which  we  have  to  fill,  multiply  that  number  by  itself,  add  i, 
again  multiply  the  number  thus  obtained  by  the  number  of  the  cells 
in  each  side,  and,  finally,  divide  the  product  by  2.  Thus,  with  4 
times  4  cells  or  squares,  we  get  :  4  times  4  are  16,  16  and  i  are  17, 
and  one  half  of  17  times  4  is  34.  Similarly,  with  5  times  5  squares, 
we  get :  5  times  5  are  25,  and  i  makes  26,  and  the  half  of  26  times 
5  is  65.  Analogously,  for  6  times  6  squares  the  summation  iii  is 
obtained,  for  7  times  7  squares  175,  for  8  times  8  squares  260,  for  9 
times  9  squares  369,  for  10  times  10  squares  505,  and  so  on.  The 
Hindu  rule  for  the  construction  of  magic  squares  whose  roots  are 
odd,  may  be  enunciated  as  follows  :  To  start  with,  write  i  in  the 
centre  of  the  topmost  row,  then  write  2  in  the  lowest  space  of  the 

=175 
=175 
=175 
=175 
=175 
=175 
=175 

175  175    175  175  175  175  175 
Fig.  5. 

vertical  column  next  adjacent  to  the  right,  and  then  so  inscribe  the 
remaining  numbers  in  their  natural  order  in  the  squares  diagonally 
upwards  towards  the  right,  that  on  reaching  the  right-hand  margin 


30 

39 

48 

I 

10 

19 

28 

38 

47 

7 

9 

18 

27 

29 

46 

6 

8 

17 

26 

35 

37 

5 

14 

16 

25 

34 

36 

45 

13 

15 

24 

33 

42 

44 

4 

21 

23 

32 

41 

43 

3 

12 

22 

31 

40 

49 

2 

II 

20 

THE  MAGIC  SQUARE. 


45 


the  inscription  shall  be  continued  from  the  left-hand  margin  in  the 
row  just  above,  and  on  reaching  the  upper  margin  shall  be  continued 
from  the  lower  margin  in  the  column  next  adjacent  to  the  right, 
noting  that  whenever  we  are  arrested  in  our  progress  by  a  square 
already  occupied  we  are  to  fill  out  the  square  next  beneath  the  one 
we  have  last  filled.  In  this  manner,  for  example,  the  last  preced- 
ing square  of  7  times  7  cells  is  formed,  in  which  the  reader  is  re- 
quested to  follow  the  numbers  in  their  natural  sequence  (Fig.  5). 

For  the  next  further  advancements  of  the  theory  of  magic 
squares  and  of  the  methods  for  theJr  construction  we  are  indebted 
to  the  Byzantian  Greek,  Moschopulus,  who  lived  in  the  fourteenth 
century  ;  also,  after  Albrecht  Diirer  who  lived  about  the  year  1500, 
to  the  celebrated  arithmetician  Adam  Riese,  and  to  the  mathemati- 
cian Michael  Stifel,  which  two  last  lived  about  1550.  In  the  seven- 
teenth century  Bachet  de  M^ziriac,  and  Athanasius  Kircher  em- 
ployed themselves  on  magic  squares.  About  1700,  finally,  the 
French  mathematicians  De  la  Hire  and  Sauveur  made  considerable 
contributions  to  the  theory.  In  recent  times  mathematicians  have 
concerned  themselves  much  less  about  magic  squares,  as  they  have 
indeed  about  mathematical  recreations  generally.  But  quite  recently 
the  Brunswick  mathematician  Scheffler  has  put  forth  his  own  and 
other's  studies  on  this  subject  in  an  elegant  form. 


5 

6 

7 
13 

M 

21 

4 

12 

20 

28 

3 

II 

19 

27 

35 

2 

10 

18 

26 

34 

42 

I 

9 

17 

25 

33 

41    49 

8 

16 

24 

32 

40 

48 

15 

23 

31 

39 

47 

22 

30 

38 

46 

29 

36 

37 

43 

44 

45 

Fig.  6. 


46 


THE  MAGIC  SQUARE. 


The  best  known  of  the  various  methods  of  constructing  magic 
squares  of  an  odd  number  of  cells  is  the  following.  First  write  the 
numbers  in  diagonal  succession  as  in  the  preceding  diagram  (Fig.  6). 
After  25  cells  of  the  square  of  49  cells  which  we  have  to  fill  out, 
have  thus  been  occupied,  transfer  the  six  figures  found  outside  each 
side  of  the  square,  without  changing  their  configuration,  into  the 
empty  cells  of  the  side  directly  opposite.  By  this  method,  which 
we  owe  to  Bachet  de  Meziriac,  we  obtain  the  following  magic  square 
of  the  numbers  from  i  to  49  : 


4 

35 
10 

41 
16 

47 
22 

29 
II 
42 

17 
48 

23 

5 

IS 

'36 

18 

49 
24 
6 

30 

37 
19 
43 
25 
7 
31 
13 

20 

44 
26 
I 

32 
14 
38 

45 
27 
2 

33 

8 

39 

21 

28 

3 
34 

9 
40 

15 

46 

Fig.  7 


MODERN  MODES  OF  CONSTRUCTION  OF  ODD-NUMBERED 
SQUARES. 

The  reader  will  justly  ask  whether  there  do  not  exist  other  cor- 
rect magic  squares  which  are  constructed  after  a  different  method 
from  that  just  given,  and  whether  there  do  not  exist  modes  of  con- 
struction which  will  lead  to  all  the  imaginable  and  possible  magic 
squares  of  a  definite  number  of  cells.  A  general  mode  of  construc- 
tion of  this  character  was  first  given  for  odd-numbered  squares  by 
De  la  Hire,  and  recently  perfected  by  Professor  Scheffler. 

To  acquaint  ourselves  with  this  general  method,  let  us  select 
as  our  example  a  square  of  5.  First  we  form  two  auxiliary  squares. 
In  the  first  we  write  the  numbers  from  i  to  5  five  times  ;  and  in  the 
second,  five  times,  the  following  multiples  of  five,  viz.:  o,  5,  10,  15, 
20.  It  is  clear  now  that  by  adding  each  of  the  numbers  of  the  series 
from  I  to  5  to  each  of  the  numbers  o,  5,  10,  15,  20,  we  shall  get 
all  the  25  numerals  from  i  to  25.  All  that  additionally  remains  to 
be  done  therefore,  is,  so  to  inscribe  the  numbers  that  by  the  addition 


THE  MAGIC  SQUARE. 


47 


of  the  two  numbers  in  any  two  corresponding  cells  each  combina- 
tion shall  come  out  once  and  only  once  ;  and  further  that  in  each 
horizontal,  vertical,  and  diagonal  row  in  each  auxiliary  square  each 
number  shall  once  appear.  Then  the  required  sum  of  65  must 
necessarily  result  in  every  case,  because  the  numbers  from  i  to  5 
added  together  make  15,  and  the  numbers  o,  5,  10,  15,  20  make  50. 
We  effect  the  required  method  of  inscription  by  imagining  the 
numbers  i,  2,  3,  4,  5  (or  o,  5,  10,  15,  20)  arranged  in  cyclical  suc- 
cession, that  is  I  immediately  following  upon  5,  and,  starting  from 
any  number  whatsoever,  by  skipping  each  time  either  none  or  one 
or  two  or  three  etc.  figures.  Cycles  are  thus  obtained  of  the  first,  the 
second,  the  third  etc.  orders  ;  for  example  3  4  5  i  2  is  a  cycle  of  the 
first  order,  241  3  5  is  a  cycle  of  the  second  order,  i  5  4  3  2  is  a 
cycle  of  the  fourth  order,  etc.  The  only  thing  then  to  be  looked  out 
for  in  the  two  auxiliary  squares  is,  that  the  same  "cycle  "  order  be 
horizontally  preserved  in  all  the  rows,  that  the  same  also  happens  for 
the  vertical  rows,  but  that  the  cycle  order  in  the  horizontal  and  ver- 
tical rows  is  different.  Finally  we  have  only  additionally  to  take 
care  that  to  the  same  numbers  of  the  one  auxiliary  square  not  like 
numbers  but  differe7it  numbers  correspond  in  the  other  auxiliary 
square,  that  is  lie  in  similarly  situated  cells.  The  following  auxiliary 
squares  are,  for  example,  thus  possible  : 


3 

4 

5 

I 

2 

0 

10 

20 

5 

15 

5 

I 

2 

3 

4 

. 

5 

15 

0 

10 

20 

2 

3 

4 

5 

I 

and 

10 

20 

5 

15 

0 

4 

5 

I 

2 

3 

15 

0 

10 

20 

5 

I 

2 

3 

4 

5 

20 

5 

15 

0 

10 

Fig.  8. 


Fig.  9. 


Adding  in  pairs  the  numbers  which  occupy  similarly  situated 
cells,  we  obtain  the  following  correct  magic  square  : 


3 

10 

12 

19 
21 

14 

16 



23 

5 
7 

25 
2 

9 
II 
18 

6 

13 
20 
22 
4 

17 
24 

I 

8 
15 

Fig.  10. 


48 


THE   MAGIC  SQUARE. 


It  will  be  seen  that  we  are  able  thus  to  construct  a  very  large 
number  of  magic  squares  of  5  times  5  spaces  by  varying  in  every 
possible  manner  the  numbers  in  the  two  auxiliary  squares.  Further- 
more, the  squares  thus  formed  possess  the  additional  peculiarity, 
that  every  5  numbers  which  fill  out  two  rows  that  are  parallel  to  a 
diagonal  and  lie  on  different  sides  of  the  diagonal  also  give  the  con- 
stant sum  of  65.  For  example:  3  and  7,  11,  20,  24;  ,or  10,  14  and  18, 
22,  I.  Altogether  then  the  sum  65  is  produced  out  of  20  rows  or 
pairs  of  rows.  On  this  peculiarity  is  dependent  the  fact  that  if  we 
imagine  an  unlimited  number  of  such  squares  placed  by  the  side  of, 
above,  or  beneath  an  initial  one,  we  shall  be  able  to  obtain  as  many 
quadratic  cells  as  we  choose,  so  arranged  that  the  square  composed 
of  any  25  of  these  cells  will  form  a  correct  magic  square,  as  the  fol- 
lowing figure  will  show  : 


2 

13 

24 

10 

16 

2 

13 

24 

10 

16 

2 

9 

20 

I 

12 

23 

9 

20 

I 

12 

23 

9 

II 

22 

8 

19 

5 

II 

22 

8 

19 

5 

II 

18 

4 
6 

15 
17 

21 

7 

18 
25 

4 
6 

15 

17 

21 

7 
14 

18 

25 

3 

14 

3 

25 

2 

13 

24 

10 

16 

2 

13 

24 

10 

16 

2 

9 

20 

I 

12 

23 

9 

20 

I 

12 

23 

9 

II 

22 

8 

19 

5 

n 

22 

8 

19 

5 

II 

18 

4 

15 

21 
3 

7 
14 

18 
25 

4 

15 
17 

21 
3 

7 

18 

25 

6 

17 

6 

14 

25 

2 

13 

24 

10 

16 

2 

13 

24 

10 

16 

2 

9 

20 
22 

I 
8 

12 
19 

23 
5 

9 
II 

20 

I 
8 

12 
19 

23 
5 

9 

II 

22 

II 

'  Fig.   II. 

Every  square  of  every  25  of  these  numbers,  as  for  example  the 
two  dark-bordered  ones,  possesses  the  property  that  the  addition  of 
the  horizontal,  vertical,  and  diagonal  rows  gives  each  the  same 
sum,  65. 

As  an  example  of  a  higher  number  of  cells  we  will  append  here 
a  magic  square  of  11  times  11  spaces  formed  by  the  general  method 
of  De  la  Hire  from  the  two  auxiliary  squares  of  Figs.  12  and  13. 
From  these  two  auxiliary  squares  we  obtain  by  the  addition  of  the 


THE  MAGIC  SQUARE. 


49 


two  numbers  of  every  two  similarly  situated  cells,  the  magic 
square,  exhibited  in  Diagram  14,  in  which  each  row  gives  the  same 
sum  671. 


I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

0 

II 

22 

33 

44 

55 

66 

77 

88 

99 

no 

3 

4 

5 

6 

7 

8 

9 

10 

II 

I 

2 

33 

44 

55 

66 

11 

88 

99 

no 

0 

II 

22 

5 

6 

7 

8 

9 

10 

11 

I 

2 

3 

4 

66 

77 

88 

99 

no 

0 

II 

22 

33 

44 

55 

7 

8 

9 

10 

II 

I 

2 

3 

4 

5 

6 

99 

no 

0 

II 

22 

33 

44 

55 

66 

77 

88 

9 

10 

II 

I 

2 

3 

4 

5 

6 

7 

8 

II 

22 

33 

44 

55 

66 

77 

88 

99 

no 

0 

II 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

44 

55 

66 

n 

88 

99 

no 

0 

II 

22 

33 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

I 

77 

88 

99 

no 

0 

II 

22 

33 

44 

55 

66 

4 

5 

6 

7 

8 

9 

10 

II 

I 

2 

3 

no 

0 

II 

22 

33 

44 

55 

66 

77 

88 

99 

6 

7 

8 

9 

10 

II 

I 

2 

3 

4 

5 

22 

33 

44 

•55 

66 

77 

88 

99 

no 

0 

II 

8 

9 

10 

II 

I 

2 

3 

4 

5 

6, 

7 

55 

66 

77 

88 

99 

no 

0 

II 

22 

33 

44 

10 

II 

I 

2 

3 

4 

5 

6 

7 

8 

9 

88 

99 

no 

0 

II 

22 

33 

44 

55 

66 

77 

Fig.  12. 


Fig.  13. 


I 

36 

71 

106 
20 

55 
79 
114 
28 

98 

13 

~^ 
83 
118 

32 
56 
91 
5 
40 

75 
no 

25 
60 

95 
9 

44 
68 

103 

52 
87 
III 

37 
72 
107 
21 

45 
80 

115 
29 

99 
2 

49 
84 
119 
33 
57 
92 
6 

41 
^ 
100 

14 

61 
96 
10 

34 
69 

104 
18 

88 
112 
26 

73 
108 
22 

81 
116 
30 
65 
89 

3 

"38" 

85 
120 

23 

Is" 

93 
7 
42 
77 

lOI 

15 
50 

9^ 
n 

35 
70 
105 
19 
54 
~^ 

113 
27 
62 

109 
12 

47 
82 
117 

31 
66 
90 
4 

39 
74 

121 
24 
59 
94 

8 

43 
67 
102 
16 

51 
86 

Fig.  14 


EVEN-NUMBERED  SQUARES. 
Of  magic  squares  having  an  even  number  of  places  we  have 
hitherto  had  to  deal  only  with  the  square  of  4.  To  construct  squares 
of  this  description  having  a  higher  even  number  of  places,  differ- 
ent and  more  complicated  methods  must  be  employed  than  for 
squares  of  odd  numbers  of  places.  However,  in  this  case  also,  as 
in  dealing  with  the  square  of  4,  we  start  with  the  natural  sequence 


50 


THE  MAGIC  SQUARE. 


of  the  numbers  and  must  then  find  the  complements  of  the  numbers 
with  respect  to  some  other  certain  number  (as  17  in  the  square  of 
4)  and  also  effect  certain  exchanges  of  the  numbers  with  one  an- 
other. To  form,  for  example,  a  magic  square  of  6  times  6  places, 
we  inscribe  in  the  12  diagonal  cells  the  numbers  that  in  the  natural 
sequence  of  inscription  fall  into  these  places,  then  in  the  remaining 
cells  the  complements  of  the  numbers  that  belong  therein  with  re- 
spect to  37,  and  finally  effect  the  following  six  exchanges,  viz.  of 
the  numbers  33  and  3,  25  and  7,  20  and  14,  18  and  13,  10  and  9, 
and  5   and  2.      In  this  way  the  following  magic  square  is  obtained. 


I 

30 
24 
13 

12 
31 

35 
8 

23 
17 
26 
2 

34 

28 

15 
21 

9 

4 

3 

27 
16 
22 
10 
33 

32 
II 

14 
20 
29 
5 

6 
7 
19 
18 

25 
1^ 

This  square  may  also  be  constructed  by  the  method  of  De  la 
Hire,  from  two  auxiliary  squares  with  the  numbers  i,  2,  3,  4,  5,  6 
and  o,  6,  12,  18,  24,  30  respectively.  In  this  case,  however,  the 
vertical  rows  of  the  one  square  and  the  horizontal  rows  of  the  other 
must  each  so  contain  two  numbers  three  times  repeated  that  the 
summation  shall  always  remain  21  and  go  respectively.  In  this 
manner  we  get  the  magic  square  last  given  above  from  the  two  fol- 
lowing auxiliary  squares : 


0 

30 

30 

0 

30 

0 

24 

6 

24 

24 

6 

6 

18 

18 

12 

12 

12 

18 

12 

12 

18 

18 

18 

12 

6 

24 

6 

6 

24 

24 

30 

0 

0 

30 

0 

30 

and 


Fig.  16. 


Fig.  17. 


It  is  to  be  noted  in  connection  with  this  example  that  here  also 
as  in  the  case  of  odd-numbered  squares,  it  is  possible  so  to  inscribe 


THE  MAGIC  SQUARE. 


51 


six  times  the  numbers  from  i  to  6  that  each  number  shall  appear  once 
and  only  once  in  each  horizontal,  vertical,  and  diagonal  row;  for 
example,  in  the  following  manner : 


123456 
2461.35 
365214 
531642 
654321 
412563 


Fig.  18. 

But  if  we  attempt  so  to  insert,  in  a  like  manner,  the  other  set  of 
numbers  o,  6,  12,  18,  24,  30  in  a  second  auxiliary  square,  that  each 
number  of  the  first  auxiliary  square  shall  stand  once  and  once  only 
in  a  corresponding  cell  with  each  number  of  the  second  square,  all 
the  attempts  we  may  make  to  fulfil  coincidently  the  last  named  con- 
dition will  result  in  failure.  It  is  therefore  necessary  to  select  aux- 
iliary squares  like  the  two  given  above.  It  is  noteworthy,  that  the 
fulfilment  of  the  second  condition  is  impossible  only  in  the  case  of 
the  square  of  6,  but  that  in  the  case  of  the  square  of  4  or  of  the 
squaie  of  8,  for  example,  two  auxiliary  squares,  such  as  the  method 
of  De  la  Hire  requires,  are  possible.  Thus,  taking  the  square  of  4 
we  get 


I 

2 

3 

4 

0 

4 

8 

12 

4 

3 

2 

I 

and 

8 

12 

0 

4 

2 

I 

4 

3 

12 

8 

4 

0 

3 

4 

I 

2 

4 

0 

12 

8 

Fig.  19, 


Fig.  20. 


The  reader  may  form  for  himself  the  magic  square  which  these 
give. 

The  existence  of  these  two  auxiliary  squares  furnishes  a  key  to 
the  solution  of  a  pretty  problem  at  cards.  If  we  replace,  namely, 
the  numbers  i,  2,  3,  4  by  the  Ace,  the  King,  the  Queen,  and  the 
Knave,  and  the  numbers  o,  4,  8,  12  by  the  four  suits,  clubs,  spades, 
hearts,  and  diamonds,  we  shall  at  once  perceive  that  it  is  possible, 
and  must  be  so  necessarily,  quadratically  to  arrange  in  such  a  man- 
ner the  four  Aces,  the  four  Kings,  the  Four  Queens,  and  the  four 


52 


THE  MAGIC  SQUARE. 


Knaves,  that  in  each  horizontal,  vertical,  and  diagonal  row,  each 
one  of  the  four  suits  and  each  one  of  the  four  denominations  shall 
appear  once  and  once  only.  The  auxiliary  squares  above  given  fur- 
nish the  appended  solution  of  this  problem  : 


CLUBS 
ACE 

SPADES 
KING 

HEARTS 
QUEEN 

DIAMONDS 
KNAVE 

HEARTS 
KNAVE 

DIAMONDS 
QUEEN 

CLUBS 
KING 

SPADES 
ACE 

DIAMONDS 
KING 

HEARTS 
ACE 

SPADES 
KNAVE 

CLUBS 
QUEEN 

SPADES 
QUEEN 

CLUBS 
KNAVE 

DIAMONDS 
ACE 

HEAKTS 
KING 

Fig.  21. 

To  fix  the  solution  of  the  problem  in  the  memory,  observe  that, 
starting  from  the  several  corners,  each  suit  and  each  denomination 
must  be  placed  in  the  spots  of  the  move  of  a  Knight.  If  we  fix  the 
positions  of  the  four  cards  of  any  one  row,  there  will  be  only  two 
possibilities  left  of  so  placing  the  other  cards  that  the  required  con- 
dition of  having  each  suit  and  each  denomination  once  and  only 
once  in  each  row  shall  be  fulfilled. 

Of  magic  squares  of  an  even  number  of  places  we  have  up  to 
this  point  examined  only  the  squares  of  4  and  of  6.  For  the  sake  of 
completeness  we  append  here  one  of  8  and  one  of  10  places.  The 
mode  of  construction  of  these  squares  is  similar  to  the  method  above 
discussed  for  the  lower  even  numbers. 


I 

48 
25 
33 
24 
16 

57 

63 
10 
18 
39 

li 
42 

50 

7 

62 
II 
19 
38 
30 
43 
51 

6 

4 
53 
45 
28 

21 

13 
60 

5 

52 
44 
29 
37 
20 
12 
61 

59 
14 
22 

35 
27 
46 
54 
3 

58 
15 
23 
34 
26 

47 
55 
2 

8 

49 
41 
32 
40 

17 

9 

"67 

Fig.  22. 


THE  MAGIC  SQUARE. 


53 


I 

90 
80 

31 
60 
50 
61 

21 
20 
91 

99 

12 

79 
69 
42 
52 
32 
29 
82 
9 

3 

88 

23 

68 

"58" 
43 
38 
73 
18 

93 

97 
14 
11 
34 
57 
47 
64 
27 
84 
4 

96 
86 

25 
66 

45 
55 

75 
15 
6 

5 

85 
26 

65 
46 

56 
35 
7^ 
16 

95 

94 
17 
74 
37 
44 
54 
67 

24 
87 
7 

8 

28 
33 
53 

48 

63 
78 
13 

92 

19 
22 
62 
49 
59 
39 
72 
89 
2 

10 
II 

71 
40 

51 
41 
70 

30 



81 

100 

Fig.  23. 

The  magic  squares  of  even  numbers  thus  constructed  are  not 
the  only  possible  ones.  On  the  contrary,  there  are  very  many  others 
possible,  which  obey  different  laws  of  formation.  It  has  been  cal- 
culated, for  example,  that  with  the  square  of  4  it  is  possible  to  con- 
struct 880,  and  with  the  square  of  6,  several  million,  different  magic 
squares.  The  number  of  odd-numbered  magic  squares  constructible 
by  the  method  of  De  la  Hire  is  also  very  great.  With  the  square  of 
7,  the  possible  constructions  amount  to  363,916,800.  With  the 
squares  of  higher  numbers  the  multitude  of  the  possibilities  increases 
in  the  same  enormous  ratio. 


V. 

MAGIC  SQUARES  WHOSE  SUMMATION  GIVES  THE  NUMBER 
OF  A  YEAR. 

The  magic  squares  which  we  have  so  far  considered  contain 
only  the  natural  numbers  from  i  upwards.  It  is  possible,  however, 
easily  to  deduce  from  a  correct  magic  square  other  squares  in  which 
a  different  law  controls  the  sequence  of  the  numbers  to  be  inscribed. 
Of  the  squares  obtained  in  this  manner,  we  shall  devote  our  atten- 
tion here  only  to  such  in  which,  although  formed  by  the  inscription 
of  successive  numbers,  the  sum  obtained  from  the  addition  of  the 
rows  is  a  determinate  number  which  we  have  fixed  upon  beforehand, 
as  the  7iumher  of  a  year.  In  such  a  case  we  have  simply  to  add  to 
the  numbers  of  the  original  square  a  determinate  number  so  to  be 
calculated,  that  the  required  sum  shall  each  time  appear.      If  this 


54 


THE   MAGIC  SQUARE. 


sum  is  divisible  by  3,  magic  squares  will  always  be  obtainable  with 
3  times  3  spaces  which  shall  give  this  sum.  In  such  a  case  we  di- 
vide the  sum  required  by  3  and  subtract  5  from  the  result  in  order 
to  obtain  the  number  which  we  have  to  add  to  each  number  of  the 
original  square.  If  the  sum  desired  is  even  but  not  divisible  by  4, 
we  must  then  subtract  from  it  34  and  take  one  fourth  of  the  result, 
to  obtain  the  number  which  in  this  case  is  to  be  added  in  each 
place.  If,  for  example,  we  wish  to  obtain  the  number  of  the  year 
1890  as  the  resulting  sum  of  each  row,  we  shall  have  to  add  to  each 
of  the  numbers  of  an  ordinary  magic  square  of  4  times  4  spaces  the 
number  464;  in  other  words,  instead  of  the  numbers  from  i  to  16  we 
have  to  insert  in  the  squares  the  numbers  from  465  to  480.  As  the 
number  of  the  year  1892  is  divisible  by  eleven,  it  must  be  pos- 
sible  to  deduce  from  the  magic  square  constructed  by  us  at  the  con- 
clusion of  Section  III  a  second  magic  square  in  which  each  row  of 
II  cells  will  give  the  number  of  the  3'ear  1892.  To  do  this,  we  sub- 
tract from  1892  the  sum  of  the  original  square,  namely  671,  and  di- 
vide the  remainder  by  11,  whereby  we  get  iii  and  thus  perceive 
that  the  numbers  from  112  to  232  are  to  be  inscribed  in  the  cells  of 


112 

124 

136 

148 

160 

172 

184 

196 

208 

220 

232 

=1892 

M7 

159 

171 

183 

195 

207 

219 

231 

122 

123 

135 

=1893 

182 

194 

206 

218 

230 

121 

133 

^34 

146 

158 

170 

=1892 

217 

229 

120 

132 

144 

145 

157 

169 

181 

193 

205 

=1892 

131 

143 

155 

156 

168 

180 

192 

204 

216 

228 

'11 

=1892 

166 

167 

179 

191 

203 

215 

227 

118 

130 

142 

154 

=  l8f2 

190 

202 

214 

226 

117 

129 

141 

153 

165 

177 

Tis 

=1892 

225 

116 

128 

140 

152 

164 

176 

188 

189 

201 

213 

=1892 

139 

151 

163 

175 

^ 

199 

200 

212 

224 

115 

127 

=1892 

174 

186 

198 

210 

211 

223 

114 

126 

138 

150 

162 

=1893 

209 

221 

222 

"3 

125 

137 

149 

161 

173 

^ 

197 

=1892 

1892  1892  1892  1892  1892  1892  1892  1892  1892  1892  1892 
Fig.  24. 

the  square  required.  We  get  in  this  way  the  preceding  square,  from 
which  o/zt"  and  iJic  same  sum,  namely  iSg2,  ean  be  obtained  44  times, 
first  from  each  of  the  1 1  horizontal  rows,  secondly  from  each  of  the 
1 1   vertical  rows,  thirdly  from  each  of  the   two  diagonal  rows,  and 


THE  MAGIC  SQUARE. 


55 


fourthly  twenty  additional  times  from  each  and  every  pair  of  any  two 
rows  that  lie  parallel  to  a  diagonal,  have  together  1 1  cells,  and  lie 
on  different  sides  of  the  diagonal,  as  for  example,  196,  122,  158,  205, 
131,  167,  214,  140,  187,  223,  149. 


CONCENTRIC  MAGIC  SQUARES. 

The  acuteness  of  mathematicians  has  also  discovered  magic 
squares  which  possess  the  peculiar  property  that  if  one  row  after  an- 
other be  taken  away  from  each  side,  the  smaller  inner  squares  re- 
maining will  still  be  magical  squares,  that  is  to  say,  all  their  rows 
when  added  will  give  the  same  sum.  It  will  be  sufficient  to  give 
two  examples  here  of  such  squares,  (the  laws  for  their  construction 
being  somewhat  more  complicated,)  of  which  the  first  has  7  times  7 
and  the  second  8  times  8  places.  The  numbers  within  each  of  the 
dark-bordered  frames  form  with  respect  to  the  centre  smaller  squares 
which  in  their  own  turn  are  magical. 


4 

5 

6 

43 

39 

38 

40 

49 

15 

16 

33 

30 

31 

I 

48 

37 

22 

27 

26 

13 

2 

47 

36 

29 

25 

21 

14 

3 

8 

18 

24 

23 

28 

32 

42 

9 

19 

34 

17 

20 

35 

41 

10 

45 

44 

7 

II 

12 

46 

I 

56 

55 

1 1 

53 

13 

14 

57 

63 

15 

47 

22 

42 

24 

45 

2 

62 

49 

25 

40 

34 

31 

16 

3 

4 

48 

28 

37 

35 

30 

17 

61 

5 

44 

39 

26 

32 

33 

21 

60 

59 

19 

38 

27 

29 

36 

46 

6 

58 

20 

18 

43 

23 

41 

50 

7 

8 

9 

10 

54 

12 

52 

51 

64 

Fig.  25. 


Fig.  26. 


In  the  first  of  these  two  squares  the  internal  square  of  3  times  3 
places  contains  the  numbers  from  21  to  29  in  such  a  manner  that 
each  row  gives  when  added  the  sum  of  75.  This  square  lies  within 
a  larger  one  of  5  times  5  spaces,  which  contains  the  numbers  from 
13  to  37  in  such  a  manner  that  each  row  gives  the  sum  of  125. 
Finally,  this  last  square  forms  part  of  a  square  of  7  times  7  places 
which  contains  the  numbers  from  i  to  49  so  that  each  row  gives  the 
sum  of  175. 

In  the  second  square  the  inner  central  square  of  4  times  4  places 
contains  the  numbers  from  25  to  40  in  such  a  manner  that  each  row 


56 


THE  MAGIC  SQUARE. 


gives  the  sum  of  130.  This  square  is  the  middle  of  a  square  of  6 
times  6  places  which  so  contains  the  numbers  from  15  to  50  that 
each  row  gives  the  sum  195.  Finally,  this  last  square  is  again  the 
middle  of  an  ordinary  magic  square  composed  of  the  numbers  from 
I  to  64. 


MAGICAL  SQUARES  WITH  MAGICAL  PARTS. 

If  we  divide  a  square  of  8  times  8  places  by  means  of  the  two 
middle  lines  parallel  to  its  sides  into  4  parts  containing  each  4  times 
4  spaces,  we  ma}^  propound  the  problem  of  so  inserting  the  numbers 
from  I  to  64  in  these  spaces  that  not  only  the  whole  shall  form  a 
magic  square,  but  also  that  each  of  the  4  parts  individually  shall  be 
magical,  that  is  to  say,  give  the  same  sum  for  each  row.  This  prob- 
lem also  has  been  successfully  solved,  as  the  following  diagram  will 
show. 


I 

4 

63 

62 

5 

8 

59 

58 

64 

61 

2 

3 

60 

57 

6 

7 

42 

43 

24 

21 

34 

35 

32 

29 

23 

22 

41 

44 

31 

30 

33 

36 

13 

16 

51 

50 

9 

12 

55 

54 

52 

49 

14 

15 

56 

53 

10 

II 

38 

39 

28 

25 

46 

47 

20 

17 

27 

26 

37 

40 

19 

18 

45 

48 

Fig.  27. 

The  4  numbers  in  each  row  of  any  one  of  the  sub-squares  here,  gives 
130;  so  that  the  sum  of  each  one  of  the  rows  of  the  large  square 
will  be  260. 

Finally,  in  further  illustration  of  this  idea,  we  will  submit  to 
the  consideration  of  our  readers  a  very  remarkable  square  of  the 
numbers  from  i  to  81.  This  square,  which  will  be  found  on  the 
following  page  (Fig.  28),  is  divided  by  parallel  lines  into  g  parts,  of 
which  each  contains  9  consecutive  numbers  that  severally  make  up 
a  magic  square  by  themselves. 

Wonderful  as  the  properties  of  this  square  may  appear,  the  law 
by  which  the  author  constructed  it  is  equally  simple.      We  have 


THE   MAGIC  SQUARE. 


57 


simply  to  regard  the  9  parts  as  the  g  cells  of  a  magic  square  of  the 
numbers  from  I  to  IX,  and  then  to  inscribe  by  the  magic  prescript 
in  the  square  designated  as  I  the  numbers  from  i  to  9,  in  the  square 


31 

36 

29 

76 

81 

74 

13 

18 

n 

30 

32 

34 

75 

77 

79 

12 

14 

16 

35 

28 

33 

80 

73 

78 

17 

10 

15 

22 

27 

20 

40 

45 

38 

58 

63 

56 

21 

23 

25 

39 

41 

43 

57 

59 

61 

26 

19 

24 

44 

37 

42 

62 

55 

60 

67 

72 

65 

4 

9 

2 

49 

54 

47 

66 

68 

70 

3 

5 

7 

48 
53 

50 
46 

52 
51 

71 

64 

69 

8 

I 

6 

Fig.  28. 

designated  as  II  the  numbers  from  10  to  18,  and  so  on.  In  this 
way  the  square  above  given  is  obtained  from  the  following  base- 
square  : 


IV 

IX 

II 

III 

V 

vir 

VIII 

I 

VI 

Fig.  29. 


MAGIC  SQUARES  THAT  INVOLVE  THE  MOVE  OF  THE 
CHESS-KNIGHT. 

What  one  of  our  readers  does  not  know  the  problems  contained 
in  the  recreation  columns  of  our  magazines,  the  requirements  of 
which  are  to  compose  into  a  verse  8  times  8  quadratically  arranged 
syllables,  of  which  every  two  successive  syllables  stand  on  spots  so 
situated  with  respect  to  each  other  that  a  chess-knight  can  move 
from  the  one  to  the  other?  If  we  replace  in  such  an  arrangement 
the  64  successive  syllables  by  the  64  numbers  from  i  to  64,  we  shall 
obtain  a  knight-problem  made  up  of  numbers.  Methods  also  exist 
indeed  for  the  construction  of  such  dispositions  of  numbers,  which 
then  form  the  foundation  of  the  construction  of  the  problems  in  the 
newspapers.  But  the  majority  of  knight-problems  of  this  class 
are  the  outcome  of  experiment  rather  than  the  product*  of  method- 


58 


THE  MAGIC  SQUARE. 


ical  creation.  If  however  it  is  a  severe  test  of  patience  to  form  a 
knight-problem  by  experiment,  it  stands  to  reason  that  it  is  a  still 
severer  trial  to  effect  at  the  same  time  the  additional  result  that  the 
64  numbers  which  form  the  knight-problem  shall  also  form  a  magic 
square. 

This  trial  of  endurance  was  undertaken  several  decades  ago,  by 
a  pensioned  Moravian  officer  named  Wenzelides,  who  was  spending 
the  last  days  of  his  life  in  the  country.  After  a  series  of  trials  which 
lasted  years  he  finally  succeeded  in  so  inscribing  in  the  64  squares 
of  the  chess-board  the  numbers  from  i  to  64  that  successive  num- 
bers, as  well  also  as  the  numbers  64  and  i,  were  always  removed 
from  one  another  in  distance  and  direction  by  the  move  of  a  knight, 
and  that  in  addition  thereto  the  summation  of  the  horizontal  and  the 
vertical  rows  always  gave  the  same  sum  260.  Ultimatelj'  he  dis- 
covered several  squares  of  this  description,  which  were  published  in 
the  Berlin  Chess  Journal.      One  of  these  is  here  appended  : 


47 
22 
II 
62 

19 
26 

35 

38 

10 

63 
46 
21 
36 
39 
18 
27 

23 

61 
12 

25 
20 

37 
34 

64 
9 
24 
45 
40 

33 

28 

17 

49 
60 

I 

8 
13 
~^ 
41 
32 

2 

5 

52 
57 
44 
29 
16 
55 

59 

50 

7 

4 

53 

14 

31' 

42 

6 

3 

58 
51 
30 
43 
54 
15 

Fig.  30. 

The  move  of  the  knight  and  the  equality  of  the  summation  of 
the  horizontal  and  vertical  rows,  therefore,  are  the  facts  to  be  noted 
here.  The  diagonal  rows  do  not  give  the  sum  260.  Perhaps  some 
one  among  our  readers  who  possesses  the  time  and  patience  will  be 
tempted  to  outdo  Wenzelides,  and  to  devise  a  numeral  knight-prob- 
lem of  this  kind  which  will  give  260  not  only  in  the  horizontal  and 
vertical  but  also  in  the  two  diagonal  rows. 


THE   MAGIC  SQUARE. 


59 


MAGICAL  POLYGONS. 

So  far  we  have  considered  only  such  extensions  of  the  idea  un- 
derlying the  construction  of  the  magic  square  in  which  the  figure  of 
the  square  was  retained.  We  may  however  contrive  extensions  of 
the  idea  in  which  instead  of  a  square,  a  rectangle,  a  triangle,  or  a 
pentagon,  and  the  like,  appear.  Without  entering  into  the  con- 
sideration of  the  methods  for  the  construction  of  such  figures,  we 
will  give  here  of  magical  polygons  simply  a  few  examples,  all  sup- 
plied by  Professor  Scheffler  : 

i)  The  numbers  from  i  to  32  admit  of  being  written  in  a  rect- 
angle of  4  X  8  in  such  a  manner  that  the  long  horizontal  rows  give 
the  sum  of  132  and  the  short  vertical  rows  the  sum  of  66  ;  thus  : 


I 
9 
24 
32 

10 

II 
30 
3 

22 

29 
12 
21 

4 

28 
20 
13 
5 

19 
27 
6 

14 

18 

7 
26 

15 

16 

25 

8 

17 

2 
31 
23 

Fig.  31. 

2)  The  numbers  from  i  to  27  admit  of  being  so  arranged  in  three 
regular  triangles  about  a  point  which  forms  a  common  centre,  that 
each  side  of  the  outermost  triangle  will  present  6  numbers  of  the 
total  summation  g6  and  each  side  of  the  middle  triangle  4  numbers 
whose  sum  is  61  ;  as  the  following  figure  shows  : 


26 


24 


27 


20       9        II       21 
16      17 

15  8 


7         13 
4  23 

19 

I  14 


25 
Fig.  32. 


6o  THE  MAGIC  SQUARE. 

3)  The  numbers  from  i  to  80  admit  of  being  formed  about  a 
point  as  common  centre  into  4  pentagons,  such  that  each  side  of 
the  first  pentagon  from  within  contains  two  numbers,  each  side  of 
the  second  pentagon  four  numbers,  each  of  the  third  six  numbers, 
and  each  side  of  the  fourth,  outermost  pentagon  eight  numbers. 
The  sum  of  the  numbers  of  each  side  of  the  second  pentagon  is  122, 
the  sum  of  those  of  each  side  of  the  third  pentagon  is  248,  and  that 
of  those  of  each  side  of  the  fourth  pentagon  254.  Furthermore,  the 
sum  of  any  four  corner  numbers  lying  in  the  same  straight  line  with 
the  centre,  is  also  the  same  ;  namely,  92. 

I 
26  54 

31  49 

10  15  80 

76  36  44  9 

50  70  72  32 


16  ^^ 

55  71  66  2^ 

45  "5  65  37 

61  g^  24  14 

30  20  17  53 


35 


40  56 

21 
69 

6 

62 
75 
77 

41 
46 


57  58  o,  79 


19 


51 


39         68 


59 

43 

64 

73 

5^ 

:3 

67 

63     18 

38 

33 

74    42 

i; 

'     28 

4         29        34  7  78        47        52  3 

Fig-  33. 

4)  The  numbers  from  i  to  73  admit  of  being  arranged  about  a 
centre,  in  which  the  number  37  is  written,  into  three  hexagons  which 
contain  respectively  3,  5,  and  7  numbers  in  each  side  and  possess 
the  following  pretty  propertie3.  Each  hexagon  always  gives  the 
same  sum,  not  only  when  the  summation  is  made  along  its  six  sides, 
but  also  when  it  is  made  along  the  six  diameters  that  join  its  corners 


THE  MAGIC  SQUARE.  6l 

and  along  the  six  that  are  constructed  at  right  angles  to  its  sides  ; 
this  sum,  for  the  first  hexagon  from  within,  is  iii,  for  the  second 
185,  and  for  the  third  259. 

I      5      6     70    60    59    58 
63  8 

62  19    53    46    22    45  9 

61  20  24  64 

2  48  31    42    38  49  57 

3  47  39  40  44  5^ 

67  51  41  37  33  23  7 

66  50  34  35  54  II 

65  25  36     32     43  26  12 

10  30  27  13 

17  29     21     28     52     55  72 

18  71 

16     69     68      4       14      15     73 
Fig.  34- 

X. 

MAGIC  CUBES. 
Several  inquirers,  particularly  Kochansky  (1686),  Sauveur 
(1710),  Hugel  (1859),  and  Scheffler  (1882),  have  extended  the  prin- 
ciple *of  the  magic  squares  of  the  plane  to  three-dimensioned  space. 
Imagine  a  cube  divided  by  planes  parallel  to  its  sides  and  equidistant 
from  one  another,  into  cubical  compartments.  The  problem  is  then, 
so  to  insert  in  these  compartments  the  successive  natural  numbers 
that  every  row  from  the  right  to  the  left,  every  row  from  the  front 
to  the  back,  every  row  from  the  top  to  the  bottom,  every  diagonal 
of  a  square,  and  every  principal  diagonal  passing  through  the  centre 
of  the  cube  shall  contain  numbers  whose  sum  is  always  the  same. 
For  3  times  3  times  3  compartments,  a  magic  cube  of  this  descrip- 
tion is  not  constructible.  For  4  times,  4  times  4  compartments  a 
cube  is  constructible  such  that  any  row  parallel  to  an  edge  of  the 
cube  and  every  principal  diagonal  give  the  sum  of  130.  To  obtain 
a  magic  cube  of  64  compartments,  imagine  the  numbers  which  be- 
long in  the  compartments  written  on  the  upper  surface  of  the  same 


62 


THE  MAGIC  SQUARE. 


and  the  numbers  then  taken  off  in  layers  of  i6  from  the  top  down- 
wards. We  obtain  thus  4  squares  of  16  cells  each,  which  together 
make  up  the  magic  cube ;  as  the  following  diagrams  will  show  : 


First  Layer 
from  the  Top. 


Second  Layer 
from  the  Top. 


Third  Layer 
from  the  Top. 


Fourth  Layer 
from  the  Top, 


I 

48 

32 

49 

63 

18 

34 

15 

62 

19 

35 

14 

4 

45 

29 

52 

60 

21 

37 

12 

6 

43 

27 

54 

7 

42 

26 

55 

57 

24 

40 

9 

56 

25 

41 

8 

10 

39 

23 

58 

II 

38 

22 

59 

53 

28 

44 

5 

13 

36 

20 

61 

51 

30 

46 

3 

50 

31 

47 

2 

16 

33 

17 

64 

The  same  sum  130  here  comes  out  not  less  than  52  times  ;  viz. 
in  the  first  place  from  the  16  rows  from  left  to  right,  secondly  from 
the  16  rows  from  the  front  to  the  back,  thirdly  from  the  16  rows 
counting  from  the  top  to  the  bottom,  and  lastly  from  the  4  rows 
which  join  each  two  opposite  corners  of  the  cube,  namely  from  the 
rows:  I,  43,  22,  64;  49,  27,  38,  16;  13,  39,  26,  52;  61,  23,  42,  4. 

For  a  cube  with  5  compartments  in  each  edge  the  arrangement 
of  the  figures  can  so  be  made  that  all  the  75  rows  parallel  to  any  and 
every  edge,  all  the  30  rows  lying  in  any  diagonal  of  a  square,  and 
all  the  4  rows  forming  any  principal  diagonal  shall  have  one  and  the 
same  summation,  315. 

Just  as  the  magic  squares  of  an  odd  number  of  cells  could  be 
formed  with  the  aid  of  two  auxiliary  squares,  so  also  odd-numbered 
magic  cubes  can  be  constructed  with  the  help  of  three  auxiliary  cubes. 

First  Layer  from  Top.  Second  Layer  from  Top.  Third  Layer  from  Top. 


121 

27 

83 

14 

70 

2 

58 

114 

45 

96 

33 

89 

20 

71 

102 

10 

61 

117 

48 

79 

36 

92 

23 

54 

no 

67 

123 

29 

85 

II 

44 

100 

I 

57 

"3 

75 

lOI 

32 

88 

19 

76 

7 

63 

ii-g 

50 

53 

109 

40 

91 

22 

84 

15 

66 

122 

28 

115 

41 

97 

3 

59 

87 

18 

74 

105 

31 

118 

49 

80 

6 

62 

24 

55 

106 

37 

93 

Fourth  Layer  from  Top. 


Lowest  Laver. 


64 

120 

46 

77 

8 

95 

21 

52 

108 

39 

98 

4 

60 

III 

42 

104 

35 

86 

17 

73 

107 

38 

94 

25 

51 

13 

69 

125 

26 

82 

16 

72 

103 

34 

90 

47 

78 

9 

65 

116 

30 

81 

12 

68 

124 

56 

112 

43 

99 

5 

THE   MAGIC  SQUARE.  63 

In  this  manner  the  preceding  magic  cube  of  5  times  5  times  5 
compartments  is  formed,  in  which,  it  may  be  additionally  noticed, 
the  middle  number  between  i  and  125,  namely  63,  is  placed  in  the 
central  compartment ;  by  which  arrangement  the  attainment  of  the 
sum  of  315  is  assured  in  the  four  principal  diagonals  and  the  30  sub- 
diagonals.  The  condition  attained  in  the  magic  squares,  that  the. 
diagonal-pairs  parallel  to  the  sub-diagonals  also  shall  give  the  sum 
315  is  not  attainable  in  this  case  but  is  so  in  the  case  of  higher  num- 
bers of  compartments. 

CONCLUSION. 

Musing  on  such  problems  as  are  the  magic  squares  is  fascinat- 
ing to  thinkers  of  a  mathematical  turn  of  mind.  We  take  de- 
light in  discovering  a  harmony  that  abides  as  an  intrinsic  quality  in 
the  forms  of  our  thought.  The  problems  of  the  magic  squares  are 
playful  puzzles,  invented  as  it  seems  for  mere  pastime  and  sport. 
But  there  is  a  deeper  problem  underlying  all  these  little  riddles,  and 
this  deeper  problem  is  of  a  sweeping  significance.  It  is  the  phil- 
osophical problem  of  the  world-order. 

The  formal  sciences  are  creations  of  the  mind.  We  build  the 
sciences  of  mathematics,  geometry,  and  algebra  with  our  conception 
of  pure  forms  which  are  abstract  ideas.  And  the  same  order  that 
prevails  in  these  mental  constructions  permeates  the  universe,  so  that 
an  old  philosopher,  overwhelmed  with  the  grandeur  of  law,  imagined 
he  heard  its  rhythm  in  a  cosmic  harmony  of  the  spheres. 


THE  FOURTH  DIMENSION. 

MATHEMATICAL  AND  SPIRITUALISTIC. 

I. 
INTRODUCTORY. 

THE  tendency  to  generalise  long  ago  led  mathematicians  to 
extend  the  notion  of  three-dimensional  space,  which  is  the 
space  of  sensible  representation,  and  to  define  aggregates  of  points, 
or  spaces,  of  more  than  three  dimensions,  with  the  view  of  employ- 
ing these  definitions  as  useful  means  of  investigation.  They  had 
no  idea  of  requiring  people  to  imagine  four-dimensional  things  and 
worlds,  and  they  were  even  still  less  remote  from  requiring  them 
to  believe  in  the  real  existence  of  a  four-dimensioned  space.  In 
the  hands  of  mathematicians  this  extension  of  the  notion  of  space 
was  a  mere  means  devised  for  the  discovery  and  expression,  by 
shorter  and  more  convenient  ways,  of  truths  applicable  to  com- 
mon geometry  and  to  algebra  operating  with  more  than  three  un- 
known quantities.  At  this  stage,  however,  the  spiritualists  came  in, 
and  coolly  took  possession  of  this  private  property  of  the  mathema- 
ticians. They  were  in  great  perplexity  as  to  where  they  should  put 
the  spirits  of  the  dead.  To  give  them  a  place  in  the  world  acces- 
sible to  our  senses  was  not  exactly  practicable.  They  were  com- 
pelled, therefore,  to  look  around  after  some  terra  incognita,  which 
should  oppose  to  the  spirit  of  research  inborn  in  humanity  an  insu- 
perable barrier.  The  abiding-place  of  the  spirits  had  perforce  to  be 
inaccessible  to  the  senses  and  full  of  mystery  to  the  mind.  This 
property  the  four-dimensioned  space  of  the  mathematicians  pos- 
sessed. With  an  intellectual  perversity  which  science  has  no  idea  of, 
these  spiritualists  boldly  asserted,   first,  that  the  whole  world  was 


THE  FOURTH   DIMENSION.  65 

situated  in  a  four-dimensioned  space  as  a  plane  might  be  situated  in 
the  space  familiar  to  us,  secondly,  that  the  spirits  of  the  dead  lived 
in  such  a  four-dimensioned  space,  thirdly,  that  these  spirits  could 
accordingly  act  upon  the  world  and,  consequently,  upon  the  human 
beings  resident  in  it,  exactly  as  we  three-dimensioned  creatures  can 
produce  effects  upon  things  that  are  two-dimensioned  ;  for  example, 
such  effects  as  that  produced  when  we  shatter  a  lamina  of  ice,  and 
so  influence  some  possibly  existing  two-dimensioned  zV<r-world. 

Since  spiritualism,  under  the  leadership  of  a  Leipsic  Professor, 
Zollner,  thus  proclaimed  the  existence  of  a  four-dimensioned  space, 
this  notion,  which  the  mathematicians  are  thoroughly  master  of, — 
for  in  all  their  operations  with  it,  though  they  have  forsaken  the 
path  of  actual  representability,  they  have  never  left  that  of  the  truth, 
— this  notion  has  also  passed  into  the  heads  of  lay  persons  who  have 
used  it  as  a  catchword,  ordinarily  without  having  any  clear  idea  of 
what  they  or  any  one  else  mean  by  it.  To  clear  up  such  ideas  and 
to  correct  the  wrong  impressions  of  cultured  people  who  have  not  a 
technical  mathematical  training,  is  the  purpose  of  the  following 
pages.  A  similar  elucidation  was  aimed  at  in  the  tracts  which 
Schlegel(Riemann,  Berlin,  1888)  and  Cranz  (Virchow-Holtzendorff's 
Sammlung,  Nos.  112  and  113)  have  published  on  the  so-called  fourth 
dimension.  Both  treatises  possess  indubitable  merits,  but  their 
methods  of  presentation  are  in  many  respects  too  concise  to  give  to 
lay  minds  a  profound  comprehension  of  the  subject.  The  author, 
accordingly,  has  been  able  to  add  to  the  reflections  which  these  ex- 
cellent treatises  offer,  a  great  deal  that  appears  to  him  necessary 
for  a  thorough  explanation  in  the  minds  of  non-mi.  thematicians  of 
the  notion  of  the  fourth  dimension. 

II. 

THE  CONCEPT  OF  DIMENSION. 

Many  text-books  of  stereometry  begin  with  the  words :  "Every 
body  has  three  dimensions,  length,  breadth,  and  thickness."  If  we 
should  ask  the  author  of  a  book  of  this  description  to  tell  us  the 
length,  breadth,  and  thickness  of  an  apple,  of  a  sponge,  or  of  a  cloud 
of  tobacco  smoke,  he  would  be  somewhat  perplexed  and  would  prob- 


66  THE  FOURTH  DIMENSION. 

ably  say,  that  the  definition  in  question  referred  to  something  dif- 
ferent. A  cubical  box,  or  some  similar  structure,  whose  angles  are 
all  right  angles  and  whose  bounding  surfaces  are  consequently  all 
rectangles  is  the  only  body  of  which  it  can  at  all  be  unmistakably 
asserted  that  there  are  three  principal  directions  distinguishable  in 
it,  of  which  any  one  can  be  called  the  length,  any  other  the  breadth, 
and  any  third  the  thickness.  We  thus  see  that  the  notions  of  length, 
breadth,  and  thickness  are  not  sufficiently  clear  and  universal  to 
enable  us  to  derive  from  them  any  idea  of  what  is  meant  when  it  is 
said  that  every  body  possesses  three  dimensions,  or  that  the  space 
of  the  world  is  three-dimensional. 

This  distinction  may  be  made  sharper  and  more  evident  by  the 
following  considerations  :  We  have,  let  us  suppose,  a  straight  line 
on  which  a  point  is  situated,  and  the  problem  is  proposed  to  deter- 
mine the  position  of  the  point  on  the  line  in  an  unequivocal  manner. 
The  simplest  way  to  solve  this  is,  to  state  how  far  the  point  is  re- 
moved in  the  one  or  the  other  direction  from  some  given  fixed  point; 
just  as  in  a  thermometer  the  position  of  the  surface  of  the  mercury 
is  given  by  a  statement  of  its  distance  in  the  direction  of  cold  or  heat 
from  a  predetermined  fixed  point — the  point  of  freezing  water.  To 
state,  therefore,  the  position  of  a  point  on  a  straight  line,  the  sole 
datum  necessary  is  a  single  number,  if  beforehand  we  have  fixed 
upon  some  standard  line,  like  the  centimetre,  and  some  definite 
point  to  which  we  give  the  value  zero,  and  have  also  previously  de- 
cided in  what  direction  from  the  zero-point,  points  must  be  situated 
whose  position  is  expressed  by  positive  numbers,  and  also  in  what 
direction  those  must  lie  whose  position  is  expressed  by  negative 
numbers.  This  last-mentioned  fact,  that  a  single  number  is  suffi- 
cient to  determine  the  place  of  a  point  in  a  straight  line,  is  the  real 
reason  why  we  attribute  to  the  straight  line  or  to  any  part  of  it  a 
single  dimension. 

More  generally,  we  call  every  totality  or  system,  of  infinitely 
numerous  things,  <?;/i?-dimensional,  in  which  one  number  is  all  that 
is  requisite  to  determine  and  mark  out  any  particular  one  of  these 
things  from  among  the  entire  totality.  Thus,  time  is  one-dimen- 
sional.    We,  as  inhabitants  of  the  earth,  have  naturally  chosen  as 


THE  FOURTH  DIMENSION.  67 

our  unit  of  time,  the  period  of  the  rotation  of  the  earth  about  its 
axis,  namely,  the  day,  or  a  definite  portion  of  a  day.  The  zero-point 
of  time  is  regarded  in  Christian  countries  as  the  year  of  the  birth  of 
Christ,  and  the  positive  direction  of  time  is  the  time  subsequent  to 
the  birth  of  Christ.  These  data  fixed,  all  that  is  necessary  to  estab- 
lish and  distinguish  any  definite  point  of  time  amid  the  infinite  to- 
tality of  all  the  points  of  time,  is  a  single  number.  Of  course  this 
number  need  not  be  a  whole  number,  but  may  be  made  up  of  the 
sum  of  a  whole  number  and  a  fraction  in  whose  numerator  and  de- 
nominator we  may  have  numbers  as  great  as  we  please.  We  may, 
therefore,  also  say  that  the  totality  of  all  conceivable  numerical 
magnitudes,  or  of  only  such  as  are  greater  than  one  definite  number 
and  smaller  than  some  other  definite  number,  is  one-dimensional. 

We  shall  add  here  a  few  additional  examples  of  one-dimen- 
sional magnitudes  presented  by  geometry.  First,  the  circumference 
of  a  circle  is  a  one-dimensional  magnitude,  as  is  every  curved  line, 
whether  it  returns  into  itself  or  not.  Further,  the  totality  of  all 
equilateral  triangles  which  stand  on  the  same  base  is  one-dimen- 
sional, or  the  totality  of  all  circles  that  can  be  described  through 
two  fixed  points.  Also,  the  totality  of  all  conceivable  cubes  will  be 
seen  to  be  one-dimensional,  provided  they  are  distinguished,  not 
with  respect  to  position,  but  with  respect  to  magnitude. 

In  conformity  with  the  fundamental  ideas  by  which  we  define 
the  notion  of  a  one-dimensional  manifoldness,  it  will  be  seen  that 
the  attribute  /a/(?-dimensional  must  be  applied  to  all  totalities  of 
things  in  which  two  numbers  are  necessary  (and  sufficient)  to  dis- 
tinguish any  determinate  individual  thing  amid  the  totality.  The 
simplest  two-dimensioned  complex  which  we  know  of  is  the  plane. 
To  determine  accurately  the  position  of  a  point  in  a  plane,  the  sim- 
plest way  is  to  take  two  axes  at  right  angles  to  each  other,  that  is, 
fixed  straight  lines,  and  then  to  specify  the  distances  by  which  the 
point  in  question  is  removed  from  each  of  these  axes. 

This  method  of  determining  the  position  of  a  point  in  a  plane 
suggested  to  the  celebrated  philosopher  and  mathematician  Des- 
cartes the  fundamental  idea  of  analytical  geometry,  a  branch  of 
mathematics  in  which  by  the  simple  artifice  of  ascribing  to  ever)' 


68  THE  FOURTH  DIMENSION. 

point  in  a  plane  two  numerical  values,  determined  by  its  distances 
from  the  two  axes  above  referred  to,  planimetrical  considerations 
are  transformed  into  algebraical.  So,  too,  all  kinds  of  curves  that 
graphically  represent  the  dependence  of  things  on  time,  make  use 
of  the  fact  that  the  totality  of  the  points  in  a  plane  is  two-dimen- 
sional. For  example,  to  represent  in  a  graphical  form  the  increase 
in  the  population  of  a  city,  we  take  a  hoiizontal  axis  to  represent 
the  time,  and  a  perpendicular  one  to  represent  the  numbers  which 
are  the  measures  of  the  population.  Any  two  lines,  then,  whose 
lengths  practical  considerations  determine,  are  taken  as  the  unit  of 
time,  which  we  may  say  is  a  year,  and  as  the  unit  of  population, 
which  we  will  say  is  one  thousand.  Some  definite  year,  say  1850, 
is  fixed  upon  as  the  zero  point.  Then,  from  all  the  equally  distant 
points  on  the  horizontal  axis,  which  points  stand  for  the  years,  we 
proceed  in  directions  parallel  to  the  other  axis,  that  is,  in  the  per- 
pendicular direction,  just  so  much  upwards  as  the  numbers  which 
stand  for  the  population  of  that  year  require.  The  terminal  points 
so  reached,  or  the  curve  which  runs  through  these  terminal  points, 
will  then  present  a  graphic  picture  of  the  rates  of  increase  of  the 
population  of  the  town  in  the  different  years.  The  rectangular  axes 
of  Descartes  are  employed  in  a  similar  way  for  the  construction  of 
barometer  curves,  which  specify  for  the  different  localities  of  a 
country  the  amount  of  variation  of  the  atmospheric  pressure  during 
any  period  of  time.  Immediately  next  to  the  plane  the  surface  of 
the  earth  will  be  recognised  as  a  two-dimensional  aggregate  of 
points.  In  this  case  geographical  latitude  and  longitude  supply  the 
two  numbers  that  are  requisite  accurately  to  determine  the  position 
of  a  point.  Also,  the  totality  of  all  the  possible  straight  lines  that 
can  be  drawn  through  any  point  in  space  is  two-dimensional,  as  we 
shall  best  understand  if  we  picture  to  ourselves  a  plane  which  is 
cut  in  a  point  by  each  of  these  straight  lines  and  then  remember 
that  by  such  a  construction  every  point  on  the  plane  will  belong  to 
some  one  line  and,  vice  versa,  a  line  to  every  point,  whence  it  fol- 
lows that  the  totality  of  all  the  straight  lines,  or,  as  we  may  call 
them,  rays,  which  pass  through  the  point  assigned  are  of  the  same 
dimensions  as  the  totality  of  the  points  of  the  imagined  plane. 


THE  FOURTH  DIMENSION.  6g 

The  question  might  be  asked,  In  what  way  and  to  what  extent 
in  this  case  is  the  specification  of  two  numbers  requisite  and  suffi- 
cient to  determine  amid  all  the  rays  which  pass  through  the  speci- 
fied point  a  definite  individual  ray?  To  get  a  clear  idea  of  the 
problem  here  involved,  let  us  imagine  the  ray  produced  far  into  the 
heavens  where  some  quite  definite  point  will  correspond  to  it*  Now, 
the  position  of  a  point  in  the  heavens  depends,  as  does  the  position 
of  a  point  on  all  spherical  surfaces,  on  two  numbers.  In  the  heavens 
these  two  numbers  are  ordinarily  supplied  by  the  two  angles  called 
altitude,  or  the  distance  above  the  plane  of  the  horizon,  and  azi- 
muth, or  the  angular  distance  between  the  circle  on  which  the  alti- 
tude is  measured  and  the  meridian  of  the  observer.  It  will  be  seen 
thus  that  the  totality  of  all  the  luminous  rays  that  an  eye,  conceived 
as  a  point,  can  receive  from  the  outer  world  is  two-dimensional,  and 
also  that  a  luminous  point  emits  a  two-dimensional  group  of  luminous 
rays.  It  will  also  be  observed,  in  connexion  with  this  example,  that 
the  two-dimensional  totality  of  all  the  rays  that  can  be  drawn  through 
a  point  in  space  is  something  different  from  the  totality  of  the  rays 
that  pass  through  a  point  but  are  required  to  lie  in  a  given  plane.  Such 
a  group  of  objects  as  the  last-named  one  is  a  one-dimensional  totality. 

Now  that  we  have  sufficiently  discussed  the  attributes  that  are 
characteristic  of  one  and  two-dimensional  aggregates,  we  may, 
without  any  further  investigation  of  the  subject,  propose  the  follow- 
ing definition,  that,  generally,  an  n- dimensional  totality  of  infinitely 
numerous  things  is  such  that  the  specification  of  n  numbers  is  necessary 
and  sufficient  to  indicate  definitely  any  individual  amid  all  the  infinitely 
numerous  individuals  of  that  totality. 

Accordingly,  the  point- aggregate  made  up  of  the  world-space 
which  we  inhabit,  is  a  three-dimensional  totality.  To  get  our  bear- 
ings in  this  space  and  to  define  any  determinate  point  in  it,  we  have 
simply  to  lay  through  any  point  which  we  take  as  our  zero-point 
three  axes  at  right  angles  to  each  other,  one  running  from  right  to 
left,  one  backwards  and  forwards,  and  one  upwards  and  downwards. 
We  then  join  each  two  of  these  axes  by  a  plane  and  are  enabled 
thus  to  specify  the  position  of  every  point  in  space  by  the  three  per- 
pendicular distances  by  which  the  point  in  question  is  removed  in  a 


70  THE  FOURTH  DIMENSION. 

positive  or  negative  sense  from  these  three  planes.  It  is  customary 
to  denote  the  numbers  which  are  the  measures  of  these  three  dis- 
tances by  X,  y,  and  z,  the  positive  x,  positive  7,  and  positive  z  ordi- 
narily being  reckoned  in  the  right  hand,  the  hitherward,  and  the 
upward  directions  from  the  origin.  If  now,  with  direct  reference  to 
this  fundamental  axial  system,  any  particular  specification  of  x,  y, 
and  z  be  made,  there  will,  by  such  an  operation,  be  cut  out  and  iso- 
lated from  the  three-dimensional  manifoldness  of  all  the  points  of 
space  a  totality  of  less  dimensions.  If,  for  example,  z  is  equal  to 
seven  units  or  measures,  this  is  equivalent  to  a  statement  that  only 
the  two-dimensional  totality  of  the  points  is  meant,  which  consti- 
tute the  plane  that  can  be  laid  at  right  angles  to  the  upward-passing 
z-axis  at  a  distance  of  seven  measures  from  the  zero-point.  Conse- 
quently, every  imaginable  equation  between  x,  y,  and  z  isolates  and 
defines  a  two-dimensional  aggregate  of  points.  If  two  different  equa- 
tions obtain  between  x,  y,  and  z,  two  such  two-dimensional  totalities 
will  be  isolated  from  among  all  the  points  of  space.  But  as  these 
last  must  have  some  one-dimensional  totality  in  common,  we  may 
say  that  the  co-existence  of  two  equations  between  x,y,  and  2  defines 
a  one-dimensional  totality  of  points,  that  is  to  say  a  straight  line,  a 
line  curved  in  a  plane,  or  even,  perhaps,  one  curved  in  space.  It 
is  evident  from  this  that  the  introduction  of  the  three  axes  of  refer- 
ence forms  a  bridge  between  the  theory  of  space  and  the  theory  of 
equations  involving  three  variable  quantities,  x,  y,  z.  The  reason 
that  the  theory  of  space  cannot  thus  be  brought  into  connection 
with  algebra  in  general,  that  is,  with  the  theory  of  indefinitely  nu- 
merous equations,  but  only  with  the  algebra  of  three  quantities,  x, 
y,  z,  is  simply  to  be  sought  in  the  fact  that  space,  as  we  picture  it, 
can  have  only  three  dimensions. 

We  have  now  only  to  supply  a  few  additional  examples  of  n- 
dimensional  totalities.  All  particles  of  air  are  four-dimensional  in 
magnitude  when  in  addition  to  their  position  in  space  we  also  con- 
sider the  variable  densities  which  they  assume,  as  they  are  expressed 
by  the  different  heights  of  the  barometer  in  the  different  parts  of  the 
atmosphere.  Similarly,  all  conceivable  spheres  in  space  are  four- 
dimensional  magnitudes,  for  their  centres  form  a  three-dimensional 


THE  FOURTH  DIMENSION.  7I 

point-aggregate,  and  around  each  centre  there  may  be  additionally 
conceived  a  one-dimensional  totality  of  spheres,  the  radii  of  which 
can  be  expressed  by  every  numerical  magnitude  from  zero  to  infinity. 
Further,  if  we  imagine  a  measuring-stick  of  invariable  length  to  as- 
sume every  conceivable  position  in  space,  the  positions  so  obtained 
will  constitute  a  five-dimensional  aggregate.  For,  in  the  first  place, 
one  of  the  extremities  of  the  measuring  stick  may  be  conceived  to 
assume  a  position  at  every  point  of  space,  and  this  determines  for 
one  extremity  alone  of  the  stick  a  three-dimensional  totality  of  po- 
sitions ;  and  secondly,  as  we  have  seen  above,  there  proceeds  from 
every  such  position  of  this  extremity  a  two-dimensional  totality  of 
directions,  and  by  conceiving  the  measuring-stick  to  be  placed 
lengthwise  in  every  one  of  these  directions  we  shall  obtain  all  the 
conceivable  positions  which  the  second  extremity  can  assume,  and 
consequently,  the  dimensions  must  be  3  plus  2  or  5.  Finally,  to 
find  out  how  many  dimensions  the  totality  of  all  the  possible  posi- 
tions of  a  square,  invariable  in  magnitude,  possesses,  we  first  give 
one  of  its  corners  all  conceivable  positions  in  space,  and  we  thus 
obtain  three  dimensions.  One  definite  point  in  space  now  being 
fixed  for  the  position  of  one  corner  of  the  square,  we  imagine  drawn 
through  this  point  all  possible  lines,  and  on  each  we  lay  off  the 
length  of  the  side  of  the  square  and  thus  obtain  two  additional  di- 
mensions. Through  the  point  obtained  for  the  position  of  the  sec- 
ond corner  of  the  square  we  must  now  conceive  all  the  possible  di- 
rections drawn  that  are  perpendicular  to  the  line  thus  fixed,  and 
we  must  lay  off  once  more  on  each  of  these  directions  the  side  of 
the  square.  By  this  last  determination  the  dimensions  are  only  in- 
creased by  one,  for  only  one  one-dimensional  totality  of  perpen- 
dicular directions  is  possible  to  one  straight  line  in  one  of  its  points. 
Three  corners  of  the  square  are  now  fixed  and  therewith  the  posi- 
tion of  the  fourth  also  is  uniquely  determined.  Accordingly,  the 
totality  of  all  equal  squares  which  differ  from  one  another  only  by 
their  position  in  space,  constitutes  a  manifoldness  of  six  dimen- 
sions. 


72  THE  FOURTH  DIMENSION. 


III. 


THE  INTRODUCTION  OF  THE  NOTION  OF  FOUR-DIMENSIONAL 
POINT-AGGREGATES,   PERMISSIBLE. 

In  the  preceding  section  it  was  shown  that  we  can  conceive  not 
only  of  manifoldnesses  of  one,  two,  and  three  dimensions,  but  also 
of  manifoldnesses  of  any  number  of  dimensions.  But  it  was  at  the 
same  time  indicated  that  our  world-space,  that  is,  the  totality  of  all 
conceivable /<7/«/'j-  that  differ  only  in  respect  of  position,  cannot  in 
agreement  with  our  notions  of  things  possess  more  than  three  di- 
mensions. But  the  question  now  arises,  whether,  if  the  progress  of 
science  tends  in  such  a  direction,  it  is  permissible  to  extend  the  no- 
tion of  space  by  the  introduction  of  point-aggregates  of  more  than 
three  dimensions,  and  to  engage  in  the  study  of  the  properties  of 
such  creations,  although  we  know  that  notwithstanding  the  fact 
that  we  may  conceptually  establish  and  explore  such  aggregates  of 
points,  yet  we  cannot  picture  to  ourselves  these  creations  as  we  do 
the  spatial  magnitudes  which  surround  us,  that  is,  the  regular  three- 
dimensional  aggregates  of  points. 

To  show  the  reader  clearly  that  this  question  must  be  answered 
in  the  aflfirmative,  that  the  extension  of  our  notion  of  space  is  per- 
missible, although  it  leads  to  things  which  we  cannot  perceive  by 
our  senses,  I  may  call  the  reader's  attention  to  the  fact  that  in  arith- 
metic we  are  accustomed  from  our  youth  upwards  to  extensions  of 
ideas,  which,  zfccurately  viewed,  as  little  admit  of  graphic  concep- 
tion as  a  four-dimensional  space,  that  is,  a  point-aggregate  of  four 
dimensions.  By  his  senses  man  first  reaches  only  the  idea  of  whole 
numbers — the  results  of  counting.  The  observation  of  primitive 
peoples*  and  of  children  clearly  proves  that  the  essential  decisive 
factors  of  counting  are  these  three  :  First,  we  abstract,  in  the  count- 
ing of  things,  completely  from  the  individual  and  characteristic  at- 
tributes of  these  things,  that  is,  we  consider  them  as  homogeneous. 
Second,  we  associate  individually  with   the  things  which  we  count 

*See  the  essay  Notion  and  Definition  of  Number  in  this  collection 


THE  FOURTH  DIMENSION  73 

other  homogeneous  things.  These  other  things  are  even  now,  among 
uncivilised  peoples,  the  ten  fingers  of  the  two  hands.  They  may, 
however,  be  simple  strokes,  or,  as  in  the  case  of  dice  and  dominoes, 
black  points  on  a  white  background.  Third,  we  substitute  for  the 
result  of  this  association  some  concise  symbol  or  word  ;  for  example, 
the  Romans  substituted  for  three  things  counted,  three  strokes  placed 
side  by  side,  namely:  III  ;  but  for  greater  numbers  of  things  they 
employed  abbreviated  signs.  The  Aztecs,  the  original  inhabitants 
of  Mexico,  had  time  enough,  it  seems,  to  express  all  the  numbers 
up  to  nineteen  by  equal  circles  placed  side  by  side.  They  had  ab- 
breviated signs  only  for  the  numbers  20,  400,  8000,  and  so  forth. 
In  speaking,  some  one  same  sound  might  be  associated  with  the 
things  counted  ;  but  this  method  of  counting  is  nowadays  employed 
only  by  clocks:  the  languages  of  men  since  prehistoric  times  have 
fashioned  concise  words  for  the  results  of  the  association  in  question. 
From  the  notion  of  number,  thus  fixed  as  the  result  of  counting,  man 
reached  the  notion  of  the  addition  of  two  numbers,  and  thence  the 
notion  that  is  the  inverse  of  the  last  process,  the  notion  of  subtrac- 
tion. But  at  this  point  it  clearly  appears  that  not  every  problem 
which  may  be  propounded  is  soluble  ;  for  there  is  no  number  which 
can  express  the  result  of  the  subtraction  of  a  number  from  one 
which  is  equally  large  or  from  one  which  is  smaller  than  itself.  The 
primary  school  pupil  who  says  that  8  from  5  "won't  go"  is  per- 
fectly right  from  his  point  of  view.  For  there  really  does  not  exist 
any  result  of  counting  which  added  to  eight  will  give  five. 

If  humanity  had  abided  by  this  point  of  view  and  had  rested 
content  with  the  opinion  that  the  problem  "5  minus  8"  is  not  solv- 
able, the  science  of  arithmetic  would  never  have  received  its  full 
development,  and  humanity  would  not  have  advanced  as  far  in  civ- 
ilisation as  it  has.  Fortunately,  men  said  to  themselves  at  this 
crisis:  "If  5  minus  8  won't  go,  we'll  make  it  go;  if  5  minus  8  does 
not  possess  an  intelligible  meaning,  we  will  simply  give  it  one."  As 
a  fact,  things  which  have  not  a  meaning  always  afford  men  a  pleas- 
ing opportunity  of  investing  them  with  one.  The  question  is,  then, 
what  significance  is  the  problem  "5  minus  8"  to  be  invested  with? 


74  THE  FOURTH   DIMENSION. 

The  most  natural  and  therefore  the  most  advantageous  solu- 
tion undoubtedly  is  to  abide  by  the  original  notion  of  subtraction  as 
the  inverse  of  addition,  and  to  make  the  significance  of  5  minus  8 
such,  that  for  5  minus  8  plus  8  we  shall  get  our  original  minuend  5. 
By  such  a  method  all  the  rules  of  computation  which  apply  to  real 
differences  will  also  hold  good  for  unreal  differences,  such  as  5  minus 
8.  But  it  then  clearly  appears  that  all  forms  expressive  of  differ- 
ences in  which  the  numbers  that  stand  before  the  minus  symbol 
are  less  by  an  equal  amount  than  those  which  follow  it  may  be  re- 
garded as  equal ;  so  that  the  simplest  course  seems  to  be  to  intro- 
duce as  the  common  characteristic  of  all  equal  differential  forms  of 
this  description  a  common  sign,  which  will  indicate  at  the  same 
time  the  difference  of  the  two  numbers  thus  associated.  Thus  it 
came  about  that  for  5  minus  8,  as  well  as  for  every  differential  form 
which  can  be  regarded  as  equal  thereto,  the  sign  "  —  3"  was  intro- 
duced. But  in  calling  differential  forms  of  this  description  num- 
bers, the  notion  of  number  was  extended  and  a  new  domain  was 
opened  up,  namely  the  domain  of  negative  numbers. 

In  the  further  development  of  the  science  of  arithmetic,  through 
the  operation  of  division  viewed  as  the  inverse  of  multiplication,  a 
second  extension  of  the  idea  of  number  was  reached,  namely,  the 
notion  of  fractional  numbers  as  the  outcome  of  divisions  that  had 
led  to  numbers  hitherto  undefined.  We  find,  thus,  that  the  science 
of  arithmetic  throughout  its  whole  development  has  strictly  adhered 
to  the  principle  of  conformity  and  consistency  and  has  invested 
every  association  of  two  numbers,  which  before  had  no  significance, 
by  the  introduction  of  new  numbers,  with  a  real  significance,  such 
that  similar  operations  in  conformity  with  exactly  the  same  rules 
could  be  performed  with  the  new  numbers,  viewed  as  the  results  of 
this  association,  as  with  the  numbers  which  were  before  known  and 
perfectly  defined.  Thus  the  science  proceeded  further  on  i*:s  way  and 
reached  the  notions  of  irrational,  imaginary,  and  complex  numbers. 

The  point  in  all  this,  which  the  reader  must  carefully  note,  is, 
that  all  the  numbers  of  arithmetic,  with  the  exception  of  he  posi- 
tive whole  numbers,  are  artificial  products  of  human  thought,  in- 
vented to  make   the  language  of  arithmetic   more  flexible,  and  to 


THE  FOURTH  DIMENSION.  75 

accelerate  the  progress  of  science.     All  these  numbers  lack  the  at- 
tributes of  representability. 

No  man  in  the  world  can  picture  to  himself  "minus  three 
trees."  It  is  possible,  of  course,  to  know  that  when  three  trees 
of  a  garden  have  been  cut  down  and  carried  away,  three  are 
missing,  and  by  substituting  for  "missing"  the  inverse  notion  of 
"added,"  we  may  say,  perhaps,  that  "minus  three  trees"  are  added. 
But  this  is  quite  different  from  the  feat  of  imagining  a  negative 
number  of  trees.  We  can  only  picture  to  ourselves  a  number  of 
trees  that  results  from  actual  counting,  that  is,  a  positive  whole 
number.  Yet,  notwithstanding  all  this,  people  had  not  the  slight- 
est hesitation  in  extending  the  notion  of  number.  Exactly  so  must 
it  be  permitted  us  in  geometry  to  extend  the  notion  of  space,  even 
though  such  an  extension  can  only  be  mentally  defined  and  can  never 
be  brought  within  the  range  of  human  powers  of  representation. 

In  mathematics,  in  fact,  the  extension  of  any  notion  is  admis- 
sible, provided  such  extension  does  not  lead  to  contradictions  with 
itself  or  with  results  which  are  well  established.  Whether  such 
extensions  are  necessary,  justifiable,  or  important  for  the  advance- 
ment of  science  is  a  different  question.  It  must  be  admitted,  there- 
fore, that  the  mathematician  is  justified  in  the  extension  of  the  no- 
tion of  space  as  a  point-aggregate  of  three  dimensions,  and  in  the 
introduction  of  space  or  point-aggregates  of  more  than  three  dimen- 
sions, and  in  the  employment  of  them  as  means  of  research.  Other 
sciences  also  operate  with  things  which  they  do  not  know  exist,  and 
which,  though  they  are  sufficiently  defined,  cannot  be  perceived  by 
our  senses.  For  example,  the  physicist  employs  the  ether  as  a 
means  of  investigation,  though  he  can  have  no  sensory  knowledge 
of  it.  The  ether  is  nothing  more  than  a  means  which  enables  us  to 
comprehend  mechanically  the  effects  known  as  action  at  a  distance 
and  to  bring  them  within  the  range  of  a  common  point  of  view. 
Without  the  assumption  of  a  material  which  penetrates  everything, 
and  by  means  of  whose  undulations  impulses  are  transmitted  to  the 
remotest  parts  of  space,  the  phenomena  of  light,  of  heat,  of  gravi- 
tation, and  of  electricity  would  be  a  jumble  of  isolated  and  uncon- 
nected mysteries.   The  assumption  of  an  ether,  however,  comprises 


76  THE  FOURTH  DIMENSION. 

in  a  systematic  scheme  all  these  isolated  events,  facilitates  our  men- 
tal control  of  the  phenomena  of  nature,  and  enables  us  to  produce 
these  phenomena  at  will.  But  it  must  not  be  forgotten  in  such  re- 
flexions that  the  ether  itself  is  even  a  greater  problem  for  man,  and 
that  the  ether-hypothesis  does  not  solve  the  difficulties  of  phenom- 
ena, but  only  puts  them  in  a  unitary  conceptual  shape.  Notwith- 
standing all  this,  physicists  have  never  had  the  least  hesitation  in 
employing  the  ether  as  a  means  of  investigation.  And  as  little  do 
reasons  exist  why  the  mathematicians  should  hesitate  to  investigate 
the  properties  of  a  four-dimensioned  point-aggregate,  with  the  view 
of  acquiring  thus  a  convenient  means  of  research. 

IV. 

THE  INTRODUCTION  OF  THE  IDEA  OF  FOUR-DIMENSIONED  POINT- 
AGGREGATES  OF  SERVICE  TO  RESEARCH. 

From  the  concession  that  the  mathematician  has  the  right  to 
define  and  investigate  the  properties  of  point-aggregates  of  more 
than  three  dimensions,  it  does  not  necessarily  follow  that  the  intro- 
duction of  an  idea  of  this  description  is  of  value  to  science.  Thus, 
for  example,  in  arithmetic,  the  introduction  of  operations  which 
spring  from  involution,  as  involution  and  its  two  inverse  operations 
proceed  from  multiplication,  is  undoubtedly  permitted.  Just  as  for 
"a  times  a  times  a"  we  write  the  abbreviated  symbol  "^z^,"  (which 
we  read,  a  to  the  third  power,)  and  investigate  in  detail  the  opera- 
tion of  involution  thus  defined,  so  we  might  also  introduce  some 
shorthand  symbol  for  "a  to  the  a"'  power  to  the  a*  power"  and  thus 
reach  an  operation  of  the  fourth  degree,  which  would  regard  a  as 
a  passive  number  and  the  number  3,  or  any  higher  number,  as  the 
active  number,  that  is,  as  the  number  which  indicates  how  often  a 
is  taken  as  the  base  of  a  power  whose  exponent  may  he  a,  or  "^  to 
the  a>^\  "  or  "  <z  to  the  a*  to  the  a'"  power. " 

But  the  introduction  of  such  an  operation  of  the  fourth  degree 
has  proved  itself  to  be  of  no  especial  value  to  mathematics.  And 
the  reason  is  that  in  the  operation  of  involution  the  law  of  commuta- 
tion does  not  hold  good.  In  addition,  the  numbers  to  be  added  may 
be  interchanged  and  the  introduction  of  multiplication  is  therefore 


THE  FOURTH  DIMENSION.  77 

of  great  value.  So,  also,  in  multiplication  the  numbers  which  are 
combined,  that  is,  the  factors,  may  be  changed  about  in  any  way, 
and  thus  the  introduction  of  involution  is  of  value.  But  in  involution 
the  base  and  the  exponent  cannot  be  interchanged,  and  consequently 
the  introduction  of  any  higher  operation  is  almost  valueless. 

But  with  the  introduction  of  the  idea  of  point-aggregates  of 
multiple  dimensions  the  case  is  wholly  different.  The  innovation 
in  question  has  proved  itself  to  be  not  only  of  great  importance  to 
research,  but  the  progress  of  science  has  irresistibly  forced  investi- 
gators to  the  introduction  of  this  idea,  as  we  shall  now  set  forth  in 
detail. 

In  the  first  place,  algebra,  especially  the  algebraical  theory  of 
systems  of  eqations,  derives  much  advantage  from  the  notion  of 
multi-dimensioned  spaces.  If  we  have  only  three  unknown  quanti- 
ties, X,  y,  z,  the  algebraical  questions  which  arise  from  the  possible 
problems  of  this  class  admit,  as  we  have  above  seen,  of  geometrical 
representation  to  the  eye.  Owing  to  this  possibility  of  geometrical 
representation,  some  certain  simple  geometrical  ideas  like  "mov- 
ing," "lying  in,"  "intersecting,"  and  so  forth,  may  be  translated 
into  algebraical  events.  Now,  no  reason  exists  why  algebra  should 
stop  at  three  variable  quantities  ;  it  must  in  fact  take  into  considera- 
tion any  number  of  variable  quantities. 

For  purposes  of  brevity  and  greater  evidentness,  therefore,  it  is 
quite  natural  to  employ  geometrical  forms  of  speech  in  the  consid- 
eration of  more  than  three  variables.  But  when  we  do  this,  we  as- 
sume, perhaps  without  really  intending  to  do  so,  the  idea  of  a  space 
of  more  than  three  dimensions.  If  we  have  four  variable  quantities, 
X,  y,  z,  u,  we  arrive,  by  conceiving  attributed  to  each  of  these  four 
quantities  every  possible  numerical  magnitude,  at  a  four-dimen- 
sioned manifoldness  of  numerical  quantities,  which  we  may  just  as 
well  regard  as  a  four-dimensioned  aggregate  of  points.  Two  equa- 
tions which  exist  on  this  supposition  between  x,  y,  z,  and  u,  define 
two  three-dimensioned  aggregates  of  points,  which  intersect,  as  we 
may  briefly  say,  in  a  two-dimensioned  aggregate  of  points,  that  is, 
in  a  surface  ;  and  so  on.  In  a  somewhat  different  manner  the  de- 
termination of  the  contents  of  a  square  or  a  cube  by  the  involution 


78  THE  FOURTH  DIMENSION. 

of  a  number  which  stands  for  the  length  of  its  sides,  leads  to  the 
notion  of  four-dimensioned  structures,  and,  consequently,  to  the 
notion  of  a  four-dimensioned  point-space.  When  we  note  that  a"^ 
stands  for  the  contents  of  a  square,  and  a^  for  the  content  of  a  cube, 
we  naturally  inquire  after  the  contents  of  a  structure  which  is  pro- 
duced from  the  cube  as  the  cube  is  produced  from  the  square  and 
which  also  will  have  the  contents  a*.  We  cannot,  it  is  true,  clearly 
picture  to  ourselves  a  structure  of  this  description,  but  we  can, 
nevertheless,  establish  its  properties  with  mathematical  exactness.* 
It  is  bounded  by  8  cubes  just  as  the  cube  is  bounded  by  6  squares  ; 
it  has  i6  corners,  24  squares,  and  32  edges,  so  that  from  every  cor- 
ner 4  edges,  6  squares,  and  4  cubes  proceed,  and  from  every  edge 
3  squares  and  3  cubes. 

Yet  despite  the  great  service  to  algebra  of  this  idea  of  multi- 
dimensioned  space,  it  must  be  conceded  that  the  conception  al- 
though convenient  is  yet  not  indispensable.  It  is  true,  algebra  is 
in  need  of  the  idea  of  multiple  dimensions,  but  it  is  not  so  abso- 
lutely in  need  of  the  idea  of /^/«/- aggregates  of  multiple  dimensions. 

This  notion  is,  however,  necessary  and  serviceable  for  a  pro- 
found comprehension  of  geometry.  The  system  of  geometrical 
knowledge  which  Euclid  of  Alexandria  created  about  three  hundred 
years  before  Christ,  supplied  during  a  period  of  more  than  two 
thousand  years  a  brilliant  example  of  a  body  of  conclusions  and 
truths  which  were  mutually  consistent  and  logical.  Up  to  the  pres- 
ent century  the  idea  of  elementary  geometry  was  indissolubly  bound 
up  with  the  name  of  Euclid,  so  that  in  England  where  people  ad- 
hered longest  to  the  rigid  deductive  system  of  the  Grecian  mathe- 
matician, the  task  of  "learning  geometry"  and  "reading  Euclid" 
were  until  a  few  years  ago  identical.  Every  proposition  of  this 
Euclidean  system  rests  on  other  propositions,  as  one  building-stone 
in  a  house  rests  upon  another.  Only  the  very  lowest  stones,  the 
foundations,  were  without  supports.      These  are  the  axioms  or  fun- 


*  Victor  Schlegel,  indeed,  has  made  models  of  the  three-dimensional  nets  of  all 
the  six  structures  which  correspond  in  four-dimensioned  space  to  the  five  regular 
bodies  of  our  space,  in  an  analogous  manner  to  that  by  which  we  draw  in  a  plane 
the  nets  of  five  regular  bodies,     Schlegel's  models  are  made  by  Brill  of  Darmstadt. 


THE  FOURTH  DIMENSION.  79 

damental  propositions,  truths  on  which  all  other  truths  are,  directly 
or  indirectly,  founded,  but  which  themselves  are  assumed  without 
demonstration  as  self-evident. 

But  the  spirit  of  mathematical  research  grew  in  time  more  and 
more  critical,  and  finally  asked,  whether  these  axioms  might  not  pos- 
sibly admit  of  demonstration.  Especially  was  a  rigid  proof  sought 
for  the  eleventh*  axiom  of  Euclid,  which  treats  of  parallels. 

After  centuries  of  fruitless  attempts  to  prove  Euclid's  eleventh 
axiom.  Gauss,  and  with  him  Bolyai  and  Lobach^vski,  Riemann, 
and  Helmholtz,  finally  stated  the  decisive  reasons  why  any  attempt 
to  prove  the  axiom  of  the  parallels  must  necessarily  be  futile.  These 
reasons  consist  of  the  fact  that  though  this  axiom  holds  good  enough 
in  the  world-space  such  as  we  do  and  can  conceive  it,  yet  three- 
dimensioned  spaces  are  ideally  conceivable  though  not  capable  of 
mental  representation,  where  the  axiom  does  not  hold  good.  The 
axiom  was  thus  shown  to  be  a  mere  fact  of  observation,  and  from  that 
time  on  there  could  no  longer  be  any  thought  of  a  deductive  demon- 
stration of  it.  In  view  of  the  intimate  connection,  which  both  in  an 
historical  and  epistemological  point  of  view  exists  between  the  ex- 
tension of  the  concept  of  space  and  the  critical  examination  of  the 
axioms  of  Euclid,  we  must  enter  at  somewhat  greater  length  into 
the  discussion  of  the  last  mentioned  propositions.  , 

Of  the  axioms  which  Euclid  lays  at  the  foundation  of  his 
geometry,  only  the  following   three  are  really  geometrical  axioms  : 

Eighth  axiom:  Magnitudes  which  coincide  with  one  another 
are  equal  to  one  another. 

Eleventh  axiom:  If  a  straight  line  meet  two  straight  lines  so  as 
to  make  the  two  interior  angles  on  the  same  side  of  it  taken  to- 
gether less  than  two  right  angles,  these  straight  lines,  being  con- 
tinually produced,  shall  at  length  meet  on  that  side  on  which  are 
the  angles  which  are  less  than  two  right  angles. 

Twelfth  axiom:  Two  straight  lines  cannot  inclose  a  [finite] 
space. 

The  numerous  proofs  which  in  the  course  of  time  were  adduced 

*AIso  called  the  twelfth  axiom,  also  the  fifth  postulate. —  Tr. 


8o  THE  FOURTH  DIMENSION. 

in  demonstration  of  these  axioms,  especially  of   the  eleventh,  all 

turn  out  on  close  examination  to  be  pseudo-proofs.    Legendre  drew 

attention  to  the  fact  that  either  of  the  following  axioms  might  be 

substituted  for  the  eleventh  : 

a)  Given  a  straight  line,  there  can  be  drawn  through  a  point 

in  the  same  plane  with  that  line,  one  and  one  line  only  which  shall 

not  intersect  the  first  (parallels)  however  far  the  two  lines  may  be 

> 
produced  ; 

d)  If  two  parallel  lines  are  cut  by  a  third  straight  line,  the  in- 
terior alternate  angles  will  be  equal. 

c)  The  sum  of  the  angles  of  a  triangle  is  equal  to  two  right 
angles,  that  is,  to  the  angle  of  a  straight  line  or  to  i8o°. 

By  the  aid  of  any  one  of  these  three  assertions,  the  eleventh 
axiom  of  Euclid  may  be  proved,  and,  vtce  versa,  by  the  aid  of  the 
latter  each  of  the  three  assertions  may  be  proved,  of  course  with 
the  help  of  the  other  two  axioms,  eight  and  twelve.  The  percep- 
tion that  the  eleventh  axiom  does  not  admit  of  demonstration  with- 
out the  employment  of  one  of  the  foregoing  substitutes  may  best 
be  gained  from  the  consideration  of  congruent  figures.  Every 
reader  will  remember  from  his  first  instruction  in  geometry  that  the 
congruence  of  two  triangles  is  demonstrated  by  the  superposition 
of  one  triangle  on  the  other  and  by  then  ascertaining  whether  the 
two  completely  coincide,  no  assumptions  being  made  in  the  deter- 
mination except  those  above  mentioned. 


a  ^^ 

V 

^^          n 

\ 

c 

Fig.  33. 

In  the  case  of  triangles  which  are  congruent,  as  are  I  and  II  in 
the  preceding  cut,  this  coincidence  may  be  effected  by  the  simple 
displacement  of  one  of  the  triangles  ;  so  that  even  a  two-dimensional 
being,  supposed  to  be  endowed  with  powers  of  reasoning,  but  only 
capable  of  picturing  to  itself  motions  within  a  plane,  also  might 
convince  itself  that*  the  two  triangles  I  and  II  could  be  made  to 
coincide.      But  a  being  of  this  description  could  not  convince  itself 


THE  FOURTH  DIMENSION.  bl 

in  like  manner  of  the  congruence  of  triangles  I  and  III.  It  would 
discover  the  equality  of  the  three  sides  and  the  three  angles,  but  it 
could  never  succeed  in  so  superposing  the  two  triangles  on  each 
other  as  to  make  them  coincide.  A  three-dimensional  being,  how- 
ever, can  do  this  very  easily.  It  has  simply  to  turn  triangle  I  about 
one  of  its  sides  and  to  shove  the  triangle,  thus  brought  into  the  po- 
sition of  its  reflexion  in  a  mirror,  into  the  position  of  triangle  III. 
Similarly,  triangles  II  and  III  may  be  made  to  coincide  by  moving 
either  out  of  the  plane  of  the  paper  around  one  of  its  sides  as  axis 
and  turning  it  until  it  again  falls  in  the  plane  of  the  paper.  The 
triangle  thus  turned  over  can  then  be  brought  into  the  position  of 
the  other. 

Later  on  we  shall  revert  to  these  two  kinds  of  congruence  : 
"congruence  by  displacement"  and  "congruence  by  circumversion." 
For  the  present  we  will  start  from  the  fact  that  it  is  always  possible 
within  the  limits  of  a  plane  to  take  a  triangle  out  of  one  position 
and  bring  it  into  another  without  altering  its  sides  and  angles.  The 
question  is,  whether  this  is  only  possible  in  the  plane,  or  whether 
it  can  also  be  done  on  other  surfaces. 

We  find  that  there  are  certain  surfaces  in  which  this  is  possi- 
ble, and  certain  others  in  which  it  is  not.  For  instance,  it  is  im- 
possible to  move  the  triangle  drawn  on  the  surface  of  an  egg  into 
some  other  position  on  the  egg's  surface  without  a  distension  or 
contraction  of  some  of  the  triangle's  parts.  On  the  other  hand,  it 
is  quite  possible  to  move  the  triangle  drawn  on  the  surface  of  a 
sphere  into  any  other  position  on  the  sphere's  surface  without  a 
distension  or  contraction  of  its  parts.  The  mathematical  reason  of 
this  fact  is,  that  the  surface  of  a  sphere,  like  the  plane,  has  every- 
where the  same  curvature,  but  that  the  surface  of  an  egg  at  differ 
ent  places  has  different  curvatures.  Of  a  plane  we  say  that  it  has 
everywhere  the  curvature  zero ;  of  the  surface  of  a  sphere  we  say  it 
has  everywhere  a  positive  curvature,  which  is  greater  in  proportion 
as  the  radius  is  smaller.  There  are  surfaces  also  which  have  a  con- 
stant negative  curvature ;  these  surfaces  exhibit  at  every  point  in 
directions  proceeding  from  the  same  side  a  partly  concave  and  a 
partly  convex   structure,   somewhat   like    the    centre    of   a   saddle. 


82  THE  FOURTH  DIMENSION. 

There  is  no  necessity  of  our  entering  in  any  detail  into  the  charac- 
ter and  structure  of  the  last-mentioned  surfaces. 

Intimately  related  with  the  plane,  however,  are  all  those  sur- 
faces, which,  like  the  plane,  have  the  curvature  zero  ;  in  this  cate- 
gory belong  especially  cylindrical  surfaces  and  conical  surfaces.  A 
sheet  of  paper  of  the  form  of  the  sector  of  a  circle  may,  for  exam- 
ple, be  readily  bent  into  the  shape  of  a  conical  surface.  If  two  con- 
gruent triangles,  now,  be  drawn  on  the  sheet  of  paper,  which  may 
by  displacement  be  translated  the  one  into  the  other,  these  triangles 
will,  it  is  plain,  also  remain  congruent  on  the  conical  surface  ;  that 
is,  on  the  conical  surface  also  we  may  displace  the  one  into  the 
other;  for  though  a  bending  of  the  figures  will  take  place,  there 
will  be  no  distension  or  contraction.  Similarly,  there  are  surfaces 
which,  like  the  sphere,  have  everywhere  a  constant  positive  curva- 
ture. On  such  surfaces  also  every  figure  can  be  transferred  into 
some  other  position  without  distension  or  contraction  of  its  parts. 
Accordingly,  on  all  surfaces  thus  related  to  the  plane  or  sphere, 
the  assumption  which  underlies  the  eighth  axiom  of  Euclid,  that  it 
is  possible  to  transfer  into  any  new  position  any  figure  drawn  on 
such  surfaces  without  distortion,  holds  good. 

The  eleventh  axiom  in  its  turn  also  holds  good  on  all  surfaces 
of  constant  curvature,  whether  the  curvature  be  zero  or  positive; 
only  in  such  instances  instead  of  "straight"  line  we  must  say 
"shortest"  line.  On  the  surface  of  a  sphere,  namely,  two  shortest 
lines,  that  is,  arcs  of  two  great  circles,  always  intersect,  no  matter 
whether  they  are  produced  in  the  direction  of  the  side  at  which  the 
third  arc  of  a  great  circle  makes  with  them  angles  less  than  two 
right  angles,  or,  in  the  direction  of  the  other  side,  where  this  arc 
makes  with  them  angles  of  more  than  two  right  angles.  On  the 
plane,  however,  two  straight  lines  intersect  only  on  the  side  where 
a  third  straight  line  that  meets  them  makes  with  them  interior  an- 
gles less  than  two  right  angles. 

The  twelfth  axiom  of  Euclid,  finally,  only  holds  good  on  the 
plane  and  on  the  surfaces  related  to  it,  but  not  on  the  sphere  or  other 
surfaces  which,  like  the  sphere,  have  a  constant  positive  curvature. 
This  also  accounts  for  the  fact    that  one  of    the  three  postulates 


THE  FOURTH  DIMENSION.  83 

which  we  regarded  as  substitutes  for  the  eleventh  axiom,  though 
vahd  for  the  plane,  is  not  true  for  the  surface  of  a  sphere;  namely, 
the  postulate  that  defines  the  sum  of  the  angles  of  a  triangle.  This 
sum  in  a  plane  triangle  is  two  right  angles  ;  in  a  spherical  tri- 
angle it  is  more  than  two  right  angles,  the  spherical  triangle  be- 
ing greater,  the  greater  the  excess  the  sum  of  its  angles  is  above 
two  right  angles.  It  will  be  seen,  from  these  considerations,  that 
in  geometries  in  which  curved  surfaces  and  not  fixed  planes  are 
studied,  the  axioms  of  Euclid  are  either  all  or  partially  false. 

The  axioms  of  geometry  thus  having  been  revealed  as  facts  of 
experience,  the  question  suggested  itself  whether  in  the  same  way 
in  which  it  was  shown  that  different  two-dimensional  geometries 
were  possible,  also  different  three-dimensional  systems  of  geometry 
might  not  be  developed  ;  and  consequently  what  the  relations  were 
in  which  these  might  stand  to  the  geometry  of  the  space  given  by 
our  senses  and  representable  to  our  mind.  As  a  fact,  a  three-dimen- 
sional geometry  can  be  developed,  which  like  the  geometry  of  the 
surface  of  an  egg  will  exclude  the  axiom  that  a  figure  or  body  can 
be  transferred  from  any  one  part  of  space  to  any  other  and  yet  re- 
main congruent  to  itself.  Of  a  three-dimensional  space  in  which 
such  a  geometry  can  be  developed  we  say,  that  it  has  no  constant 
measure  of  curvature. 

The  space  which  is  representable  to  us,  and  which  we  shall 
henceforth  call  the  space  of  experience,  possesses,  as  our  experiences 
without  exception  confirm,  the  especial  property  that  every  bodily 
thing  can  be  transferred  from  any  one  part  of  it  to  any  other  with- 
out suffering  in  the  transference  any  distension  or  any  contraction. 
The  space  of  experience,  therefore,  has  a  constant  measure  of  cur- 
vature. The  question,  however,  whether  this  measure  of  curvature 
is  zero  or  positive,  that  is,  whether  the  space  of  experience  possesses 
the  properties  which  in  two-dimensional  structures  a  plane  pos- 
sesses, or  whether  it  is  the  three  dimensional  analogue  of  the  surface 
of  a  sphere  is  one  which  future  experience  alone  can  answer.  If  the 
space  of  experience  has  a  constant  positive  measure  of  curvature 
which  is  different  from  zero,  be  the  difference  ever  so  slight,  a  point 
which  should  move  forever  onward  in  a  straight  line,  or,  more  ac- 


84  THE  FOURTH  DIMENSION. 

curately  expressed,  in  a  shortest  line,  would  sometime,  though  per 
haps  after  having  traversed  a  distance  which  to  us  is  inconceivable, 
ultimately  have  to  arrive  from  the  opposite  direction  at  the  place 
from  which  it  set  out,  just  as  a  point  which  moves  forever  onward 
in  the  same  direction  on  the  surface  of  a  sphere  must  ultimately  ar- 
rive at  its  starting  point,  the  distance  it  traverses  being  longer  the 
greater  the  radius  of  the  sphere  or  the  smaller  its  curvature. 

It  will  seem,  at  first  blush,  almost  incredible,  that  the  space  of 
experience  possibh-  could  have  this  property.  But  an  example, 
which  is  the  historical  analogue  of  this  modem  transformation  of  our 
conceptions,  will  render  the  idea  less  marvellous.  Let  us  transport 
ourselves  to  the  age  of  Homer.  At  that  time  people  believed  that 
the  earth  was  a  great  disc  surrounded  on  all  sides  by  oceans  which 
were  conceived  to  be  in  all  directions  infinitely  great.  Indeed,  for 
the  primitive  man,  who  has  never  journeyed  far  from  the  place  of 
his  birth,  this  is  the  most  natural  conception.  But  imagine  now  that 
some  scholar  had  come,  and  had  informed  the  Homeric  hero  Ulysses 
that  if  he  would  travel  forever  on  the  earth  in  the  same  direction  he 
would  ultimately  come  back  to  the  point  from  which  he  started  ; 
surely  Ulysses  would  have  gazed  with  as  much  astonishment  upon 
this  scholar  as  we  now  look  upon  the  mathematician  who  tells  us 
that  it  is  possible  that  a  point  which  moves  forever  onward  in  space 
in  the  same  direction  may  ultimately  arrive  at  the  place  from  which 
it  started.  But  despite  the  fact  that  Ulysses  would  have  regarded 
the  assertion  of  the  scholar  as  false  because  contradictory  to  his 
familiar  conceptions,  that  scholar,  nevertheless,  would  have  been 
right ;  for  the  earth  is  not  a  plane  but  a  spherical  surface.  So  also 
the  mathematician  may  be  right  who  bases  this  more  recent  strange 
view  on  the  possible  fact  that  the  space  of  experience  may  have  a 
measure  of  curvature  which  is  not  exactly  zero  but  slightly  greater 
than  zero.  If  this  were  really  the  case,  the  vohane  of  the  space  of 
experience,  though  very  large,  would,  nevertheless,  be  finite  ;  just 
as  the  real  spherical  surface  of  the  earth  as  contrasted  with  the 
Homeric  plane  surface  is  finite,  having  so  and  so  many  square  miles. 
When  the  objection  is  here  made  that  a  finiteness  of  space  is  totally 
at  variance  with  our  modes  of  thought  and  conceptions,  two  ideas, 


THE  FOURTH  DIMENSION.  85 

*' infinitely  great"  and  "unlimited,"  are  confounded.  All  that  is  at 
variance  with  our  practical  conceptions  is  that  space  cian  anywhere 
have  a  boundary  ;  not  that  it  may  possibly  be  of  tremendous  but 
finite  magnitude. 

It  will  now  be  asked  if  we  cannot  determine  by  actual  observa- 
tion whether  the  measure  of  curvature  of  experiential  space  is  ex- 
actly zero  or  slightly  different  therefrom.  The  theorem  of  the  sum 
of  the  angles  of  a  triangle  and  the  conclusions  which  follow  from 
this  theorem  do  indeed  supply  us  with  a  means  of  ascertaining  this 
fact.  And  the  results  of  observation  have  been,  that  the  measure  oj 
curvature  of  space  is  in  all  probability  exactly  equal  to  zero  or  if  it  is 
slightly  different  from  zero  it  is  so  little  so  that  the  technical  means  of 
observation  at  our  conunand  and  especially  our  telescopes  are  not  compe- 
tent to  determine  the  a7noutit  of  the  deviation.  More,  we  cannot  with 
any  certainty  say. 

All  these  reflections,  to  which  the  criticism  of  the  hypotheses 
that  underlie  geometry  long  ago  led  investigators,  compel  us  to  in- 
stitute a  comparison  between  the  space  of  experience  and  other 
three-dimensional  aggregates  of  points  (spaces),  which  we  cannot 
mentally  represent  but  can  in  thought  and  word  accurately  define 
and  investigate.  As  soon,  however,  as  we  are  fully  implicated  in  the 
task  of  accurately  investigating  the  properties  of  three-dimensional 
aggregates  of  points,  we  find  ourselves  similarly  forced  to  regard 
such  aggregates  as  the  component  elements  of  a  manifoldness  of 
more  than  three  dimensions.  In  this  way  the  exact  criticism  of 
even  ordinary  geometry  leads  us  to  the  abstract  assumption  of  a 
space  of  more  than  three  dimensions.  And  as  the  extension  of  every 
idea  gives  a  clearer  and  more  translucent  form  to  the  idea  as  it  orig- 
inally stood,  here  too  the  idea  of  multi-dimensioned  aggregates 
of  points  and  the  investigation  of  their  properties  has  thrown  a  new 
light  on  the  truths  of  ordinary  geometry  and  placed  its  properties 
in  clearer  relief.  Among  the  numerous  examples  which  show  how 
the  notion  of  a  space  of  multiple  dimensions  has  been  of  great  ser- 
vice to  science  in  the  investigation  of  three-dimensional  space,  we 
shall  give  one  a  place  here  which  is  within  the  comprehension  of 
non-mathematicians. 


86 


THE   FOURTH  DIMENSION. 


Imagine  in  a  plane  two  triangles  whose  angles  are  denoted 
by  pairs  of  numbers — namely,  by  1-2,  1-3,  1-4,  and  2-5,  3-5,  4-5. 
(See  Fig.  36.)  Let  the  two  triangles  so  lie  that  the  three  lines  which 
join  the  angles  1-2  and  2-5,  1-3  and  3-5,  and  1-4  and  4-5  intersect  at 
a  point,  which  we  will  call  1-5.  If  now  we  cause  the  sides  of  the 
triangles  which  are  opposite  to  these  angles  to  intersect,  it  will  be 
found  that  the  points  of  intersection  so  obtained  possess  the  peculiar 
property  of  lying  all  in  one  and  the  same  straight  line.  The  point 
of  intersection  of  the  connection  1-3  and  1-4  with  the  connection  4-5 
and  3-5  may  appropriately  be  called  3-4.  Similarly,  the  point  of  in- 
tersection 2-4  is  produced 
by  the  meeting  of  4-5,  2-5 
and  1-2,  1-4;  and  the  point 
of  intersection  2-3,  by  the 
meeting  of  1-3,  1-2,  and 
3-5,  2-5.  The  statement, 
that  the  three  points  of 
intersection  3-4,  2-4,  2-3, 
thus  obtained,  lie  in  one 
straight  line,  can  be 
proved  by  the  principles 
11  of  plane  geometry  only 
with  difficulty  and  great 
circumstantiality.  But  by 
resorting  to  the  three- 
dimensional  space  of  ex- 
perience, in  which  the  plane  of  the  drawing  lies,  the  proposition 
can  be  rendered  almost  self-evident. 

To  begin  with,  imagine  any  five  points  in  space  which  may  be 
denoted  by  the  numbers  i,  2,  3,  4,  5  ;  then  imagine  all  the  possible 
ten  straight  lines  of  junction  drawn  between  each  two  of  these  points, 
namely,  1-2,  1-3  ....  4-5;  and  finally,  also,  all  the  ten  planes  of 
junction  of  every  three  points  described,  namely,  the  plane  1-2-3, 
1-2-4,  ....  3-4-5.  A  spatial  figure  will  thus  be  obtained,  whose  ten 
straight  lines  will  meet  some  interposed  plane  in  ten  points  whose 
relative  positions  are  exactly  those  of  the  ten  points  above  described. 


Fig.  36. 


THE  FOURTH  DIMENSION.  87 

Thus,  for  example,  on  this  plane  the  points  1-2,  1-3.  and  2-3  will  lie  in 
a  straight  line,  for  through  the  three  spatial  points  i,  2,  3,  a  plane  can 
be  drawn  which  will  cut  the  plane  of  a  drawing  in  a  straight  line. 
The  reason,  therefore,  that  the  three  points  3-4,  2-4,  2-3,  also  must 
ultimately  lie  in  a  straight  line,  consists  in  the  simple  fact  that  the 
plane  of  the  three  points  2,  3,  4,  must  cut  the  plane  of  the  drawing 
in  a  straight  line.  The  figure  here  considered  consists  of  ten  points 
of  which  sets  of  three  so  lie  ten  times  in  a  straight  line  that  con- 
versely from  every  point  also  three  straight  lines  proceed. 

Now,  just  as  this  figure  is  a  section  of  a  complete  three-dimen- 
sional pentagon,  so  another  remarkable  figure,  of  similar  proper- 


Fig.  37- 

ties,  may  be  obtained  from  the  section  of  a  figure  of  four-dimen- 
sioned space.  Imagine  six  points,  i,  2,  3,  4,  5,  6,  situated  in 
this  four-dimensioned  space,  and  every  three  of  them  connected  by 
a  plane,  and  every  four  of  them  by  a  three-dimensioned  space.  We 
shall  obtain  thus  twenty  planes  and  fifteen  three-dimensioned  spaces 
which  will  cut  the  plane  in  which  the  figure  is  to  be  produced  in 
twenty  points  and  fifteen  rays  which  so  lie  that  each  point  sends  out 
three  rays  and  every  ray  contains  four  points.  (See  Fig.  37.)  Fig- 
ures of  this  description,  which  are  so  composed  of  points  and  rays 
that  an  equal  number  of  rays  proceed  from  every  point  and  an  equal 


OO  THE  FOURTH  DIMENSION. 

number  of  points  lie  in  every  ray,  are  called  configurations.  Other 
configurations  may,  of  course,  be  produced,  by  taking  a  different 
number  of  points  and  by  assuming  that  the  points  taken  lie  in  a 
space  of  different  or  even  higher  dimensions.  The  author  of  this 
article  was  the  first  to  draw  attention  to  configurations  derived  from 
spaces  of  higher  dimensions.  As  we  see,  then,  the  notion  of  a  space 
of  more  than  three  dimensions  has  performed  an  important  ser- 
vice in  the  investigations  of  common  plane  geometry. 

In  conclusion,  I  should  like  to  add  a  remark  which  Cranz  makes 
regarding  the  application  of  the  idea  of  multi-dimensioned  space 
to  theoretical  chemistry.  (See  the  treatise  before  cited.)  In  chem- 
istry, the  molecules  of  a  compound  body  are  said  to  consist  of  the 
atoms  of  the  elements  which  are  contained  in  the  body,  and  these  are 
supposed  to  be  situated  at  certain  distances  from  one  another,  and 
to  be  held  in  their  relative  positions  by  certain  forces.  At  first,  the 
centres  of  the  atoms  were  conceived  to  lie  in  one  and  the  same 
plane.  But  Wislicenus  was  led  by  researches  in  paralactic  acid 
to  explain  the  differences  of  isomeric  molecules  of  the  same  struc- 
tural formulae  by  the  different  positions  of  the  atoms  in  space.  (Com- 
pare La  chimie  dans  Vespace  by  van't  Hoff,  1875,  preface  by  J. 
Wislicenus).  In  fact  four  points  can  always  be  so  arranged  in  space 
that  every  two  of  them  may  have  any  distance  from  each  other  ; 
and  the  change  of  one  of  the  six  distances  does  not  necessarily  in- 
volve the  alteration  of  any  other. 

But  suppose  our  molecule  consists  of  five  atoms?  Four  of  these 
ma}'^  be  so  placed  that  the  distance  between  any  two  of  them  can  be 
made  what  we  please.  But  it  is  no  longer  possible  to  give  the  fifth 
atom  a  position  such  that  each  of  the  four  distances  by  which  it  is 
separated  from  the  other  atoms  may  be  what  we  please.  On  the 
contrary,  the  fourth  distance  is  dependent  on  the  three  remaining 
distances  ;  for  the  space  of  experience  has  only  three  dimensions. 
If,  therefore,  I  have  a  molecule  which  consists  of  five  atoms  I  can- 
not alter  the  distance  between  two  of  them  without  at  least  altering 
some  second  distance.  But  if  we  imagine  the  centres  of  the  atoms 
placed  in  a  four-dimensioned  space,  this  can  be  done  ;  all  the  ten 
distances  which  may  be  conceived  to  exist  between  the  five  points 


THE   FOURTH  DIMENSION.  89 

will  then  be  independent  of  one  another.  To  reach  the  same  result 
in  the  case  of  six  atoms  we  must  assume  a  five-dimensional  space  ; 
and  so  on. 

Now,  if  the  independence  of  all  the  possible  distances  between 
the  atoms  of  a  molecule  is  absolutely  required  by  theoretical  chem- 
ical research,  the  science  is  really  compelled,  if  it  deals  with  mole- 
cules of  more  than  four  atoms,  to  make  use  of  the  idea  of  a  space  of 
more  than  three  dimensions.  This  idea  is,  in  this  case,  simply  an 
instrument  of  research,  just  as  are,  also,  the  ideas  of  molecules  and 
atoms- — means  designed  to  embrace  in  a  perspicuous  and  systematic 
form  the  phenomena  of  chemistry  and  to  discover  the  conditions 
under  which  new  phenomena  can  be  evoked.  Whether  a  four- 
dimensioned  space  really  exists  is  a  question  whose  insolubility 
cannot  prevent  research  from  making  use  of  the  idea,  exactly  as 
chemistry  has  not  been  prevented  from  making  use  of  the  notion 
of  atom,  although  no  one  really  knows  whether  the  things  we  call 
atoms  exist  or  not. 

V. 

REFUTATION    OF    THE    ARGUMENTS    ADDUCED    TO    PROVE    THE 

EXISTENCE  OF  A  FOUR  DIMENSIONED  SPACE  INCLUSIVE 

OF  THE  VISIBLE  WORLD. 

The  considerations  of  the  preceding  section  will  have  convinced 
the  cultured  non-mathematician  of  the  service  which  the  theory  of 
multi-dimensioned  spaces  has  done,  and  bids  fair  to  do,  for  geo- 
metrical research.  In  addition  thereto  is  the  consideration  that 
every  extension  of  one  branch  of  mathematical  science  is  a  constant 
source  of  beneficial  and  helpful  influence  to  the  other  branches.  The 
knowledge,  however,  that  mathematicians  can  employ  the  notion  of 
four-dimensioned  space  with  good  results  in  their  researches,  would 
never  have  been  sufficient  to  procure  it  its  present  popularity;  for 
every  man  of  intelligence  has  now  heard  of  it,  and,  in  jest  or  in 
earnest,  often  speaks  of  it.  The  knowledge  of  a  four-dimensioned 
space  did  not  reach  the  ears  of  cultured  non-mathematicians  until 
the  consequences  which  the  spiritualists  fancied  it  was  permissible 
to  draw  from  this  mathematical  notion  were  publicly  known.  But 
it  is  a  tremendous  step  from   the  four-dimensioned  space  of  the 


go  THE  FOURTH  DIMENSION. 

mathematicians  to  the  space  from  which  the  spirit-friends  of  the 
spiritualistic  mediums  entertain  us  with  rappings,  knockings,  and 
bad  EngHsh.  Before  taking  this  step  we  will  first  discuss  the  ques- 
tion of  the  real  existence  of  a  four-dimensional  space,  not  deciding 
the  question  whether  this  space,  if  it  really  does  exist,  is  inhabited 
by  reasonable  beings  who  consciously  act  upon  the  world  in  which 
we  exist. 

Among  the  reasons  which  are  put  forward  to  prove  the  exist- 
ence of  a  four-dimensional  space  containing  the  world,  the  least  rep- 
rehensible are  those  which  are  based  on  the  existence  of  symmetry. 
We  spoke  above  of  two  triangles  in  the  same  plane  which  have  all 


Fig.  38. 

their  sides  and  angles  congruent,  but  which  cannot  be  made  to  co- 
incide by  simple  displacement  within  the  plane  ;  but  we  saw  that 
this  coincidence  could  be  effected  by  holding  fast  one  side  of  one 
triangle  and  moving  it  out  of  its  plane  until  it  had  been  so  far  turned 
round  that  it  fell  back  into  its  plane.  Now  something  similar  to 
this  exists  in  space.  Cut  two  figures,  exactly  like  that  of  Fig.  38,  out 
of  a  piece  of  paper,  and  turn  the  triangle  ABF  about  the  side  AB, 
ACE  about  the  side  AC,  BCD  about  the  side  BC,  and  in  one  figure 
above  and  in  the  other  belowj  then  in  both  cases  the  points  B,E,B 
will  meet  at  a  point,  because  AE  is  equal  to  AF,  BF  is  equal  to 
BD,  CD  is  equal  to  CE.  In  this  manner  we  obtain  two  pyramids 
which  in  all  lengths  and  all  angles  are  congruent,  yet  which  cannot. 


THE   FOURTH  DIMENSION.  QI 

no  matter  how  we  try,  be  made  to  coincide,  that  is,  be  so  fitted  the 
one  into  the  other  that  they  shall  both  stand  as  one  pyramid.  But 
the  reflected  image  of  the  one  could  be  brought  into  coincidence 
with  the  other.  Two  spatial  structures  whose  sides  and  angles  are 
thus  equal  to  each  other,  and  of  which  each  may  be  viewed  as  the 
reflected  image  of  the  other,  are  called  symmetrical.  For  instance, 
the  right  and  the  left  hand  are  symmetrical ;  or,  a  right  and  a  left 
glove.  Now  just  as  in  two  dimensions  it  is  impossible  by  simple 
displacement  to  bring  into  congruence  triangles  which  like  those 
above  mentioned  can  only  be  made  to  coincide  by  circumversion, 
so  also  in  three  dimensions  it  is  impossible  to  bring  into  congruence 
two  symmetrical  pyramids.  Careful  mathematical  reflection,  how- 
ever, declares  that  this  could  be  effected,  if  it  were  possible,  while 
holding  one  of  the  surfaces,  to  move  the  pyramid  out  of  the  space 
of  experience,  and  to  turn  it  round  through  a  four-dimensioned 
space  until  it  reached  a  point  at  which  it  would  return  again  into 
our  experiential  space.  This  process  would  simply  be  the  four- 
dimensional  analogue  of  the  three-dimensional  circumversion  in 
the  above-mentioned  case  of  the  two  triangles.  Further,  the  interior 
surfaces  in  this  process  would  be  converted  into  exterior  surfaces, 
and  %nce  versa,  exactly  as  in  the  circumversion  of  a  triangle  the  an- 
terior and  posterior  sides  are  interchanged.  If  the  structure  which 
is  to  be  converted  into  its  symmetrical  counterpart  is  made  of  a  flex- 
ible material,  the  interchange  mentioned  of  the  interior  and  exterior 
surfaces  may  be  effected  by  simply  turning  the  structure  inside  out ; 
for  example,  a  right  glove  may  thus  be  converted  into  a  left  glove. 
Now  from  this  truth,  that  every  structure  can  be  converted,  by 
means  of  a  four-dimensional  space  inclusive  of  the  world,  into  a 
structure  symmetrical  with  it,  it  has  been  sought  to  establish  the 
probability  of  the  real  existence  of  a  four-dimensioned  space.  Yet 
it  will  be  evident,  from  the  discussion  of  the  preceding  section,  that 
the  only  inference  which  we  can  here  make  is,  that  the  idea  of  a 
four-dimensioned  space  is  competent,  from  a  mathematical  point  of 
view,  to  throw  some  light  upon  the  phenomena  of  symmetry.  To 
conclude  from  these  facts  that  a  space  of  this  kind  really  exists, 
would  be  as  daring  as  to  conclude  from  the  fact  that  the  uniform 


92  THE   FOURTH  DIMENSION. 

angular  velocity  of  the  apparent  motions  of  the  fixed  stars  is  expli- 
cable from  the  assumption  of  an  axial  motion  of  the  firmament,  that 
the  fixed  stars  are  really  rigidly  placed  in  a  celestial  sphere  rotating 
about  its  axis.  It  must  not  be  forgotten  that  our  comprehension  of 
the  phenomena  of  the  real  world  consists  of  two  elements  :  first,  of 
that  which  the  things  really  are ;  and,  second,  of  that  by  which  we 
rationally  apprehend  the  things.  This  latter  element  is  partly  de- 
pendent on  the  sum  of  the  experiences  which  we  have  before  ac- 
quired, and  partly  on  the  necessity,  due  to  the  imperfection  of  rea- 
son, of  our  classifying  the  multitudinous  isolated  phenomena  of  the 
world  into  categories  which  we  ourselves  have  formed,  and  which, 
therefore,  are  not  wholly  derived  from  the  phenomena  themselves, 
but  to  a  great  extent  are  dependent  on  us. 

Besides  geometrical  reasons,  Zollner  has  also  adduced  cosmo- 
logical  reasons  to  prove  the  existence  of  a  four-dimensional  space. 
To  these  reasons  belong  especially  the  questions,  whether  the  num- 
ber of  the  fixed  stars  is  infinitely  great,  whether  the  world  is  finite 
or  infinite  in  extension,  whether  the  world  had  a  beginning  or  will 
have  an  end,  whether  the  world  is  not  hastening  towards  a  condition 
of  equilibrium  or  dead  level  by  the  universal  distribution  of  its  matter 
and  energy  ;  the  problems,  also,  of  gravitation  and  action  at  a  dis- 
tance ;  and  finally,  the  questions  concerning  the  relations  between 
the  phenomena  in  the  world  of  sense-perception  to  the  unknown 
things-in-themselves.  All  these  questions  which  can  be  decided  in 
no  definite  sense,  led  Zollner  and  his  followers  to  the  assumption 
that  a  four-dimensioned  space  inclusive  of  the  space  of  experience 
must  really  exist.  But  more  careful  reflection  will  show  that  this 
assumption  does  not  dispose  of  the  difficulties  but  simply  displaces 
them  into  another  realm.  Furthermore,  even  if  four-dimensioned 
space  did  unravel  and  make  clear  all  the  cosmological  problems 
which  have  bothered  the  human  mind,  still,  its  existence  would  not 
be  proved  thereby;  it  would  yet  remain  a  mere  hypothesis,  designed 
to  render  more  intelligible  to  a  being  who  can  only  make  experiences 
in  a  three-dimensional  space,  the  phenomena  therein  which  are  full 
of  mystery  to  it.  A  four-dimensioned  space  would  in  such  case  pos- 
sess for  the  metaphysician  a  value  similar  to  that  which  the  ether 


THE  FOURTH  DIMENSION.  93 

possesses  for  the  physicist.  Still  more  convincing  than  these  cos- 
mological  reasons  to  the  majority  of  men  is  the  physio-psychological 
reason  drawn  from  the  phenomena  of  vision  which  Zollner  adduces. 
Into  this  main  argument  we  will  enter  in  more  detail. 

When  we  "  see  "  an  object,  as  we  all  know,  the  light  which  pro- 
ceeds or  is  reflected  therefrom  produces  an  image  on  the  retina  of 
our  eye;  this  image  is  conducted  to  our  consciousness  by  means  of 
the  optic  nerve,  and  our  reason  draws  therefrom  an  inference  re- 
specting the  object.  When,  now,  we  look  at  a  square  whose  sides 
are  a  decimetre  in  length,  and  whose  centre  is  situated  at  the  distance 
of  a  metre  from  the  pupil  of  our  eye,  an  image  is  produced  on  the 
retina.  But  exactly  the  same  image  will  be  produced  there  if  we 
look  at  a  square  whose  sides  are  parallel  to  the  sides  of  the  first 
square  but  two  decimetres  in  length,  and  whose  centre  is  situated 
at  a  distance  of  two  metres  from  the  pupil  of  the  eye.  Proceeding 
thus  further,  we  readily  discover  that  an  eye  can  perceive  in  any 
length  or  line  only  the  ratio  of  its  magnitude  to  the  distance  at  which 
it  is  situated  from  it,  and  that  generally  a  three-dimensional  world 
must  appear  to  the  eye  two-dimensional,  because  all  points  which 
lie  behind  each  other  in  the  direction  outwards  from  the  eye  pro- 
duce on  the  retina  only  one  image.  This  is  due  to  the  fact  that  the 
retinal  images  are  themselves  two-dimensional ;  for  which  reason, 
Zollner  says,  the  world  must  appear  to  a  child  as  two-dimensional, 
if  it  be  supposed  to  live  in  a  primitive  condition  of  unconscious  men- 
tal activity.  To  such  a  child  two  objects  which  are  moving  the  one 
behind  the  other,  must  appear  as  suffering  displacement  on  a  sur- 
face, which  we  conceive  behind  the  objects,  and  on  which  the  latter 
are  projected.  In  all  these  apparent  displacements,  coincidences 
and  changes  of  form  also  are  effected.  All  these  things  must  appear 
puzzling  to  a  human  being  in  the  first  stages  of  its  development, 
and  the  mind  thus  finds  itself,  as  Zollner  further  argues,  in  the  first 
years  of  childhood  forced  to  adopt  a  hypothesis  concerning  the  con- 
stitution of  space  and  to  assume  that  the  world  is  three-dimensional, 
although  the  eye  can  really  perceive  it  as  only  two-dimensional. 
Zollner  then  further  says,  that  in  the  explanation  of  the  effects  of 
the  external  world,  man  constantly  finds  this  hypothesis  of  his  child-- 


94  'i'HE  FOURTH  DIMENSION. 

ish  years  confirmed,  and  that  in  this  way  it  has  become  in  his  mind 
so  profound  a  conviction  that  it  is  no  longer  possible  for  him  to 
think  it  away.  Consonant  with  this  argumentation,  also,  is  ZoU- 
ner's  remark,  that  the  same  phenomenon  has  presented  itself  in 
astronomical  methods  of  knowledge.  To  explain  the  movements  of 
the  planets,  which  appear  to  describe  regular  paths  on  the  surface 
of  a  celestial  sphere,  we  were  compelled  in  the  solution  of  the  rid- 
dles which  these  motions  presented,  to  assume  in  the  structure  of  the 
heavens  a  dimension  of  "depth,"  and  the  complicated  motions  in 
the  two-dimensioned  firmament  were  converted  into  very  simple 
motions  in  three-dimensioned  space.  Zollner  also  contends  that  our 
conception  of  the  entire  visible  world  as  possessed  of  three  dimen- 
sions is  a  product  of  our  reason,  which  the  mind  was  driven  to  form 
by  the  contradictions  which  would  be  presented  to  it  on  the  assump- 
tion of  only  two  dimensions  by  the  perspective  distortions,  coinci- 
dences, and  changes  of  magnitude  of  objects.  When  a  child  moves 
its  hand  before  its  eyes,  turns  it,  brings  it  nearer,  or  pushes  it  farther 
away,  this  child  successively  receives  the  most  various  impressions 
on  the  surface  of  its  retina  of  one  and  the  same  object  of  whose 
identity  and  constancy  its  feelings  offer  it  a  perfect  assurance.  If 
the  child  regarded  the  changeable  projection  of  the  hand  on  the  sur- 
face of  the  retina  as  the  real  object,  and  not  the  hand  which  lies  be- 
5'ond  it,  the  child  would  constantly  be  met  with  contradictions  in  its 
experience,  and  to  avoid  this  it  makes  the  hypothesis  that  the  space 
of  experience  is  three-dimensional.  Zollner's  contention  is,  there- 
fore, that  man  originally  had  only  a  two-dimensional  intuition  of 
space,  but  was  forced  by  experience  to  represent  to  himself  the  ob- 
jects which  on  the  retinal  surface  appeared  two-dimensional,  as 
three-dimensional,  and  thus  to  transform  his  two-dimensional  space- 
intuition  into  a  three-dimensional  one.  Now,  in  exactly  the  same 
way,  according  to  Zollner's  notion,  will  man,  by  the  advancement 
and  increasing  exactness  of  his  knowledge  of  the  phenomena  of  the 
outer  world,  also  be  compelled  to  conceive  of  the  material  world  as 
a  "shadow  cast  by  a  more  real  four-dimensional  world,"  so  that 
these  conceptions  will  be  just  as  trivial  for  the  people  of  the  twen- 
tieth century  as  since  Copernicus's  time  the  explanation  of  the  mo- 


THE  FOURTH  DIMENSION.  95 

tions  of  the  heavenly  bodies  by  means  of  a  three-dimensional  mo- 
tion has  been. 

Zollner's  arguments  from  the  phenomena  of  vision  may  be  re- 
futed as  follows  :  In  the  first  place  it  is  incorrect  to  say  that  we  see 
the  things  of  the  external  world  by  means  of  two-dimensional  retinal 
images.  The  light  which  penetrates  the  eye  causes  an  irritation 
of  the  optic  nerves,  and  any  such  effect  which,  though  it  be  not 
powerful,  is,  nevertheless,  a  mechanical  one,  can  only  take  place  on 
things  which  are  material.  But  material  things  are  always  three- 
dimensional.  The  effect  of  light  on  the  sensitive  plates  of  photog- 
raphy can  with  just  as  little  justice  be  regarded  as  two-dimensional. 
Our  senses  can  have  perception  of  nothing  but  three-dimensional 
things,  and  this  perception  is  effected  by  forces  which  in  their  turn 
act  on  three-dimensional  things,  namely  our  sensory  nerves.  It  is 
wrong  to  call  an  image  two-dimensional,  for  it  is  only  by  abstraction 
that  we  can  conceive  of  a  thickness  so  growing  constantly  smaller 
and  smaller  as  to  admit  of  our  regarding  a  three-dimensional  picture 
as  two-dimensional,  by  giving  it  in  mind  a  vanishingly  small  thick- 
ness. It  is  also  wrong  to  say,  as  Zollner  says,  that  when  we  see  the 
shadow  of  a  hand  which  is  cast  upon  a  wall  we  see  something  two- 
dimensional.  What  we  really  perceive  is  that  no  light  falls  upon 
our  eye  from  the  region  included  by  the  shadow,  while  from  the 
entire  surrounding  region  light  does  fall  on  our  eye.  But  this  light 
is  reflected  from  the  material  particles  which  form  the  surface  of  the 
wall,  that  is,  from  three-dimensional  particles  of  matter.  We  must 
always  remember  that  our  eye  communicates  to  us  only  three-dimen- 
sional knowledge,  and  that  for  the  comprehension  of  anything  which 
has  two,  one,  or  no  dimensions,  a  purely  intellectual  act  of  abstraction 
must  be  added  to  the  act  of  perception.  When  we  imagine  we  have 
made  a  lead-pencil  mark  on  paper,  we  have,  exactly  viewed,  simply 
heaped  alongside  of  each  other  little  particles  of  graphite  in  such  a 
manner  that  there  are  by  far  fewer  graphite  particles  in  the  lateral 
and  upward  directions  than  there  are  in  the  longitudinal  direction, 
and  thus  our  reason  arrives  by  abstraction  at  the  notion  of  a  straight 
line.  When  we  look  at  an  object,  say  a  cube  of  wood,  we  recognise 
the  object  as  three-dimensional,  and  it  is  only  by  abstraction  that 


96  THE  FOURTH  DIMENSION. 

we  can  conceive  of  its  two-dimensional  surfaces,  of  its  twelve  one- 
dimensional  edges,  and  of  its  eight  no-dimensional  corners.  For  we 
reach  the  perception  of  its  surface,  for  example,  solely  by  reason  of 
the  fact  that  the  material  particles  which  form  the  cube  prevent  the 
transmission  of  light,  and  reflect  it,  whereby  a  part  of  the  light  re- 
flected from  every  material  particle  strikes  our  eye.  Now,  by  think- 
ing exclusivel}'  of  those  material  particles  which  are  reflected,  in 
contrariety  to  the  empty  space  without  and  the  hidden  and  there- 
fore non-reflected  particles  within,  we  form  the  notion  of  a  surface. 
It  is  evident  from  this,  that  all  that  we  perceive  is  three-dimen- 
sional, that  we  cannot  reach  anything  two-dimensional  without 
an  intellectual  abstraction,  and  that,  therefore,  we  cannot  conceive 
of  anything  two-dimensional  exerting  effects  upon  material  things. 
But  this  fact  is  a  refutation  of  the  retinal  argument  of  Zollner.  If 
vision  consisted  wholly  and  exclusively  in  the  creation  of  a  two- 
dimensional  image,  the  things  which  take  place  in  the  world  could 
never  come  into  our  consciousness.  The  child,  therefore,  does  not 
originally  apprehend  the  world,  as  Zollner  says,  as  two-dimensional; 
on  the  contrary,  it  apprehends  it  either  not  at  all,  or  it  apprehends 
it  as  three-dimensional.  Of  course  the  child  must  first  "learn  how" 
to  see.  It  is  found  from  the  observation  of  children  during  the  first 
months  of  their  lives,  and  of  the  congenitally  blind  who  have  sud- 
denly acquired  the  power  of  vision  by  some  successful  operation, 
that  seeing  does  not  consist  alone  in  the  irritations  which  arise  in 
the  optic  nerves,  but  also  in  the  correct  interpretation  of  these  irri- 
tations by  reason.  This  correct  interpretation,  however,  can  be 
accomplished  only  by  the  accumulation  of  a  considerable  stock  of 
experience.  Especially  must  the  recognition  of  the  distance  of  the 
object  seen  be  gradually  learned.  In  this,  two  things  are  especially 
helpful ;  first,  the  fact  that  we  have  two  eyes  and,  consequently, 
that  we  must  feel  two  irritations  of  the  optic  nerves  which  are  not 
wholly  alike  ;  and,  secondly,  the  fact  that  we  are  enabled  by  our 
power  of  motion  and  our  sense  of  touch  to  convince  ourselves  of 
the  distance  and  form  of  the  bodies  seen.  The  question  now  arises, 
what  sort  of  an  intuition  of  space  would  a  creature  have  that  had 
onl}'  one  eye,  that  could  neither  move  itself  nor  its  eye,  and  also 


THE  FOURTH  DIMENSION.  97 

possessed  no  peripheral  nerves.  According  to  Zollner's  view,  this 
creature  could,  owing  to  its  two-dimensional  retinal  images,  have 
only  a  two-dimensional  intuition  of  space.  The  author's  opinion, 
however,  is,  that  such  a  creature  could  not  see  at  all,  as  it  has  no 
possibility  of  collecting  experiences  which  are  adapted  in  any  way 
to  interpreting  the  effects  of  things  on  its  retina.  The  light  which 
proceeded  from  the  objects  roundabout  and  fell  on  the  retina  could 
produce  no  other  effect  on  the  being  than  that  of  a  wholly  unintel- 
ligible irritation,  or  perhaps  even  pain. 

The  reflections  presented  sufficiently  show  that  neither  the 
phenomena  of  symmetry  nor  the  retinal  images  of  the  objects  of 
vision  necessarily  force  upon  us  the  assumption  of  a  four-dimen- 
sioned space.  If  the  material  world  should  ever  present  problems 
which  could  not  in  the  progress  of  knowledge  be  solved  in  a  natural 
way,  the  assumption  that  a  four-dimensional  space  containing  the 
world  exists  would  also  be  incompetent  to  resolve  the  difficulties 
presented ;  it  would  simply  convert  these  difficulties  into  others, 
and  not  dispose  of  the  problems  but  simply  displace  them  to  an- 
other world.  Yet  the  question  might  be  asked,  is  the  existence  of 
a  four-dimensional  space  really  impossible?  To  answer  this  question 
we  must  first  clearly  know  what  we  mean  by  "exist."  If  existence 
means  that  the  intellectual  idea  of  a  thing  can  be  formed  and  that 
this  idea  shall  not  lead  to  contradictions  with  other  well-established 
ideas  and  with  experience,  we  have  only  to  say  that  four-dimen- 
sioned space  does  exist,  as  the  arguments  adduced  in  sections  III 
and  IV  hav,e  rendered  plain.  If,  namely,  the  space  of  four  dimen- 
sions did  not  exist  as  a  clear  idea  in  the  minds  of  mathematicians, 
mathematicians  could  certainly  not  have  been  led  by  this  idea  to 
results  which  are  recognised  by  the  senses  as  true,  and  which  really 
take  place  in  our  own  representable  space.  But  if  existence  means 
"material  actuality,"  we  must  say  that  we  neither  now  nor  in  the 
future  can  know  anything  about  it.  For  we  know  material  actual- 
ity only  as  three-dimensional,  our  senses  can  only  make  three- 
dimensional  experiences,  and  the  inferences  of  our  reason,  although 
they  can  well  abstract  from  material  things,  can  never  ascend  to 
the  point  of  explaining  a  four-dimensional  materiality.   Just  as  little, 


gS  THE  FOURTH  DIMENSION. 

therefore,  as  we  can  locally  fix  the  idea  of  a  two-dimensional  mate- 
rial world,  as  little  can  we  ever  verify  the  notion  of  a  four-dimen- 
sional material  existence. 


EXAMINATION  OF  THE  HYPOTHESIS  CONCERNING  THE  EXISTENCE 
OF  FOUR-DIMENSIONAL  SPIRITS. 

In  connection  with  the  belief  that  the  visible  world  is  contained 
in  a  four-dimensioned  space,  Zollner  and  his  adherents  further  hold 
that  this  higher  space  is  inhabited  by  intelligent  beings  who  can 
act  consciously  and  at  will  on  the  human  beings  who  live  in  expe- 
riential space.  To  invest  this  opinion  with  greater  strength,  Zollner 
appealed  to  the  fact  that  the  greatest  thinkers  of  antiquity  and  of 
modern  times  were  either  wholly  of  this  opinion  or  at  least  held 
views  from  which  his  contentions  might  be  immediately  derived. 
Plato's  dialogue  between  Socrates  and  Glaukon  in  the  seventh  book 
of  the  Republic,  is  evidence,  says  Zollner,  that  this  greatest  philos- 
opher of  antiquity  possessed  some  presentiment  of  this  extension  of 
the  notion  of  space.  Yet  any  one  who  has  connectedly  studied  and 
understood  Plato's  system  of  philosophy  must  concede  that  the  so- 
called  "ideas"  of  the  Platonic  system  denote  something  wholly  dif- 
ferent from  what  Zollner  sees  in  them  or  pretends  to  see.  Zollner 
says  that  these  Platonic  ideas  are  spatial  objects  of  more  than  three 
dimensions  and  represent  "real  existence"  in  the  same  sense  that 
the  material  world,  as  contrasted  with  the  images  on  the  retina,  rep- 
resents it.  Zollner  similarly  deals  with  the  Kantian  "  thing-in-it- 
self,"  which  is  also  regarded  as  an  object  of  higher  dimensions. 

To  show  Kant  in  the  light  of  a  predecessor,  Zollner  quotes  the 
following  passage  from  the  former's  "Traume  eines  Geistersehers, 
erliiutert  durch  Traume  der  Metaphysik  "  (1766,  Collected  Works, 
Vol.  VII.  page  32  et  seq.):  "I  confess  that  I  am  very  much  in- 
"  clined  to  assert  the  existence  of  immaterial  beings  in  the  world, 
*'  and  to  rank  my  own  soul  as  one  of  such  a  class.  It  appears,  there 
"is  a  spiritual  essence  existent  which  is  intimately  bound  up  with 
"matter  but  which  does  not  act  on  those  forces  of  the  elements  by 
"which  the  latter  are  connected,  but  upon  some  internal  principle 


THE  FOURTH  DIMENSION.  gg 

"of  its  own  condition.  It  will,  in  the  time  to  come — I  know  not 
"when  or  where — be  proved,  that  the  human  soul,  even  in  this  life, 
"exists  in  a  state  of  uninterrupted  connection  with  all  the  imma- 
"terial  natures  of  the  spiritual  world ;  that  it  alternately  acts  on 
"them  and  receives  impressions  from  them,  of  which,  as  a  human 
''  soul,  it  is  not,  in  the  normal  state  of  things,  conscious.  It  would 
"  be  a  great  thing,  if  some  such  systematic  constitution  of  the  spirit- 
^*ual  world,  as  we  conceive  it,  could  be  deduced,  not  exclusively 
"from  our  general  notion  of  spiritual  nature,  which  is  altogether  too 
■"hypothetical,  but  from  some  real  and  universally  admitted  ob- 
"servations, — or,  for  that  matter,  if  it  could  even  be  shown  to  be 
"probable." 

What  Kant  really  asserts  here  is,  first,  the  partly  independent 
and  partly  dependent  existence  of  the  soul,  and  of  spiritual  beings 
generally,  on  matter,  and,  second,  that  spiritual  beings  have  some 
common  connection  with  and  mutually  influence  one  another.  This 
contention,  which  is  that  of  very  many  thinkers,  does  not,  how- 
ever, entail  the  consequence  that  the  "transcendental  subject  of 
Kant"  must  be  four-dimensional,  as  Zollner  asserts  it  does.  Kant 
never  even  hinted  at  the  theory  that  the  psychical  features  of  the 
world  owe  their  connection  with  the  material  features  to  the  fact 
that  they  are  four-dimensional  and,  therefore,  include  the  three- 
dimensional.  Is  it  a  necessary  conclusion  that  if  a  thing  exists  and 
is  not  three-dimensional,  as  is  the  case  with  the  soul,  it  is  there- 
fore four-dimensional?  Can  it  not  in  fact  be  so  constituted  that  it 
is  wholly  meaningless  to  speak  of  dimensions  at  all  in  connection 
w^ith  it  ? 

Yet  still  more  strangely  than  the  words  of  Plato  and  Kant  do 
certain  utterances  of  the  mathematicians  Gauss  and  Riemann  speak 
in  favor  of  Zollner's  hypothesis.  S.  v.  Waltershausen  relates  of 
Gauss  in  his  Gruss  zuni  Geddchtniss,  (Leipsic,  1856),  that  Gauss 
had  often  remarked  that  the  three  dimensions  of  space  were  only 
a  specific  pecuh'arity  of  the  human  mind.  We  can  think  ourselves, 
he  said,  into  beings  who  are  conscious  of  only  two  dimensions  ; 
similarly,  perhaps,  beings  who  are  above  and  outside  our  world  may 
look  down  upon  us  ;  and  there  were,  he  continued,  in  a  jesting  tone, 


THE  FOURTH  DIMENSION. 


a  number  of  problems  which  he  had  here  indefinitely  laid  aside,  but 
hoped  to  treat  in  a  superior  state  by  superior  geometrical  methods. 
Leaving  aside  this  jest,  which  quite  naturally  suggested  itself,  the 
remarks  of  Gauss  are  quite  correct.  We  possess  the  power  to  ab- 
stract and  can  think,  therefore,  what  kind  of  geometry  a  being  that 
is  only  acquainted  with  a  two-dimensional  world  would  have  ;  for 
instance,  we  can  imagine  that  such  a  being  could  not  conceive  of 
the  possibility  of  making  two  triangles  coincide  which  were  con- 
gruent in  the  sense  above  explained,  and  so  on.  So,  also,  we  can 
understand  that  a  being  who  has  control  of  four  dimensions  can  only 
conceive  of  a  geometry  of  four-dimensional  space,  yet  may  have  the 
capacity  of  thinking  itself  into  spaces  of  other  dimensions.  But  it 
does  not  follow  from  this  that  a  four-dimensional  space  exists,  let 
alone  that  it  is  inhabited  by  reasonable  beings. 

Riemann,  on  the  other  hand,  speaks  directly  of  a  world  of  spir- 
its. In  his  Neue  mathematische  Principien  der  Naturphilosophie  he 
puts  forth  the  hypothesis  that  the  space  of  the  world  is  filled  with 
a  material  that  is  constantly  pouring  into  the  ponderable  atoms, 
there  to  disappear  from  the  phenomenal  world.  In  every  ponderable 
atom,  he  says,  at  every  moment  of  time,  there  enters  and  appears  a 
determinate  amount  of  matter,  proportional  to  the  force  of  gravita- 
tion. The  ponderable  bodies,  according  to  this  theor}',  are  the 
place  at  which  the  spiritual  world  enters  and  acts  on  the  material 
world.  Riemann's  world  of  spirits,  the  sole  office  of  which  is  to  ex- 
plain the  phenomenon  of  gravitation  as  a  force  governing  matter, 
is,  however,  essentially  different  from  the  spiritual  world  of  Zollner, 
the  function  of  which  is  to  explain  supposed  supersensuous  phe- 
nomena which  stand  in  the  most  glaring  contradiction  with  the  es- 
tablished known  laws  of  the  material  world. 

Besides  this  appeal  to  the  testimony  of  eminent  men  like  Plato, 
Kant,  Gauss,  and  Riemann,  the  scientific  prophet  of  modern  spirit- 
ualism also  bases  his  theory  on  the  belief,  which  has  obtained  at  all 
times  and  appeared  in  various  forms  among  all  peo'ples,  that  there 
exist  in  the  world  forces  which  at  times  are  competent  to  evoke 
phenomena  that  are  exempt  from  the  ordinary  laws  of  nature.  We 
'  have  but  to  think  of  the  phenomena  of  table-turning  which  once  ex- 


THE  FOURTH  DIMENSION.  lOI 

cited  the  Chinese  as  much  as  it  has  aroused,  during  the  last  few 
decades,  the  European  and  American  worlds  ;  or  of  the  divining- 
rod,  by  whose  help  our  forefathers  sought  for  water,  in  fact,  as  we 
do  now  in  parts  of  Europe  and  America. 

Cranz,  in  his  essay  on  the  subject,  divides  spiritualistic  phe- 
nomena into  physical  and  intellectual.  Of  the  first  class  he  enume- 
rates the  following  :  the  moving  of  chairs  and  tables  ;  the  animation 
of  walking-sticks,  slippers,  and  broomsticks;  the  miraculous  throw- 
ing of  objects ;  spirit-rappings  (Luther  heard  a  sound  in  the  Wart- 
burg,  "  as  if  three  score  casks  were  hurled  down  the  stairs  ")  ;  the 
ecstatic  suspension  of  persons  above  the  floor  ;  the  diminution  of 
the  forces  of  gravity  ;  the  ordeals  of  witches  ;  the  fetching  of  wished- 
for  objects  ;  the  declination  of  the  magnetic  needle  by  persons  at  a 
distance  ;  the  untying  of  knots  in  a  closed  string  ;  insensibility  to 
injury  and  exemption  therefrom  when  tortured,  as  in  handling  red- 
hot  coals,  carrying  hot  irons,  etc.;  the  music  of  invisible  spirits  ; 
the  materialisations  of  spirits  or  of  individual  parts  of  spirits  (the 
footprints  in  the  experiments  of  Slade,  photographed  by  Zollner)  ; 
the  double  appearance  of  the  same  person  ;  the  penetration  of  mat- 
ter (of  closed  doors,  windows,  and  so  forth).  As  numerous  also  is 
the  selection  presented  by  Cranz  of  intellectual  phenomena,  namely: 
spirit-writing  (Have's  instrument  for  the  facilitation  of  intercourse 
with  spirits),  the  clairvoyance  and  divination  of  somnambulists,  of 
visionary,  ecstatic,  and  hypnotised  persons,  prompted  or  controlled 
by  narcotic  medicines,  by  sleeping  in  temples,  by  music  and  dancing, 
by  ascetic  modes  of  life  and  residence  in  barren  localities,  by  the  ex- 
udations of  the  soil  and  of  water,  by  the  contemplation  of  jewels, 
mirrors,  and  crystal-pure  water,  and  by  anointing  the  finger-nails 
with  consecrated  oil.  Also  the  following  additional  intellectual  phe- 
nomena are  cited  :  increased  eloquence  or  suddenly  acquired  power 
of  speaking  in  foreign  languages;  spirit-effects  at  a  distance;  in- 
ability to  move,  transferences  of  the  will,  and  so  forth. 

All  these  phenomena,  presented  with  the  aspect  of  truth,  and 
associated  more  or  less  with  trickery,  self-deception,  and  humbug, 
are  adduced  by  the  spiritualists  to  substantiate  the  belief  in  a  world 
of  spirits  which  intentionally  and  consciously  take  part  in  the  events 


I02  THE   FOURTH  DIMENSION. 

of  the  material  world,  and  to  prove  that  these  phenomena  may  be 
sufficiently  and  consistently  explained  by  the  effects  of  the  activity 
of  such  a  world.  It  is  impossible  for  us  to  discuss  and  put  to  the 
test  here  the  explanations  of  all  these  supersensuous  phenomena. 
Anything  and  everything  can  be  explained  by  spirits  who  act  at  will 
upon  the  world.  There  are  only  a  few  of  these  phenomena,  namely, 
clairvoyance  and  Slade's  experiments,  whose  explanations  are  so 
intimately  connected  with  our  main  theme,  the  so-called  fourth  di- 
mension, that  they  cannot  be  passed  over. 

First,  with  respect  to  clairvoyance,  the  American  visionary  Da- 
vis describes  the  experiences  which  he  claims  to  have  made  in  this 
condition,  induced  by  "  magnetic  sleep,"  as  follows  :*  "  The  sphere 
of  my  vision  now  began  to  expand.  At  first,  I  could  only  clearly 
discern  the  walls  of  the  house.  At  the  start  they  seemed  to  me  dark 
and  gloomy;  but  they  soon  became  brighter  and  finally  transparent. 
I  could  now  see  the  objects,  the  utensils,  and  the  persons  in  the  ad- 
joining house  as  easily  as  those  in  the  room  in  which  I  sat.  But  my 
perceptions  extended  further  still ;  before  my  wandering  glance, 
which  seemed  to  control  a  great  semi-circle,  the  broad  surface  of 
the  earth,  for  hundreds  of  miles  about  me,  grew  as  transparent  as 
water,  and  I  saw  the  brains,  the  entrails,  and  the  entire  anatomy  of 
the  beasts  that  wandered  about  in  the  forests  of  the  Eastern  Hemi- 
sphere, hundreds  and  thousands  of  miles  from  the  room  in  which  I 
sat."  The  belief  in  the  possibility  of  such  states  of  clairvoyance  is 
by  no  means  new.  Alexander  Dumas  made  use  of  it,  for  example, 
in  his  Mimoires  iVun  mddicin,  in  which  Count  Balsamo,  afterwards 
called  Cagliostro,  is  said  to  possess  the  power  to  throw  suitable 
persons  into  this  wonderful  condition  and  thus  to  find  out  what 
other  persons  at  distant  places  are  doing.  Zollner  explains  clair- 
voyance by  means  of  the  fourth  dimension  thus  : 

A  man  who  is  accustomed  to  viewing  things  on  a  plain  is  sup- 
posed to  ascend  to  a  considerable  height  in  a  balloon.  He  will 
there  enjoy  a  much  more  extended  prospect  than  if  he  had  remained 
on  the  plain  below,  and  will  also  be  able  to  signal  to  greater  dis- 

*  Quoted  by  Cranz. 


THE  FOURTH  DIMENSION.  IO3 

tances.  The  plain,  that  is,  the  two-dimensioned  space,  is  accord- 
ingly viewed  by  him  from  points  outside  of  the  plain  as  "open"  in 
all  directions.  Exactly  so,  in  Zollner's  theory,  must  three-dimen- 
sioned spaces  appear,  when  viewed  from  points  in  four-dimensioned 
space,  namely,  as  "open";  and  the  more  so  in  proportion  as  the 
point  in  question  is  situated  at  a  greater  distance  from  the  place  of 
our  body,  or  in  proportion  as  the  soul  ascends  to  a  greater  height  in 
this  fourth  dimension.  Zollner  thus  explains  clairvoyance  as  a  con- 
dition in  which  the  soul  has  displaced  itself  out  of  its  three-dimen- 
sioned space  and  reached  a  point  which  with  respect  to  this  space 
is  four-dimensionally  situated  and  whence  it  is  able  to  contemplate 
the  three-dimensional  world  without  the  interference  of  intervening 
obstacles. 

The  following  remark  is  to  be  made  to  this  explanation.  The 
reason  why  we  have  a  better  and  more  extended  view  from  a  bal- 
loon than  from  places  on  the  earth  is  simply  this,  that  between  the 
suspended  balloon  and  the  objects  seen  at  a  distance  nothing  inter- 
venes but  the  air,  and  air  allows  the  transmission  of  light,  whereas, 
at  the  places  below  on  the  earth  there  are  all  kinds  of  material 
things  about  the  observer  which  prevent  the  transmission  of  light 
and  either  render  difficult  or  absolutely  impossible  the  sight  of 
things  which  lie  far  away.  In  the  same  way,  also,  from  a  point  in 
four-dimensioned  space,  a  three-dimensional  object  will  be  visible 
only  provided  there  are  no  obstacles  intervening.  If,  therefore,  this 
awareness  of  a  distant  object  is  a  real,  actual  vision  by  means  of  a 
luminous  ray  which  strikes  the  eye,  there  is  contained  in  the  ex- 
planation of  Zollner  the  tacit  assumption  that  the  medium  with 
which  the  four-dimensional  world  is  filled  is  also  pervious  to  light 
exactly  as  the  atmosphere  is. 

The  theory  that  there  are  four-dimensional  spirits  who  produce 
the  phenomena  cited  by  the  spiritualists  received  special  support 
from  the  experiments  which  the  prestidigitateur  Slade,  who  claimed 
he  was  a  spiritualistic  medium,  performed  before  Zollner.  Of  these 
experiments  we  will  speak  of  the  two  most  important,  the  experi- 
ment with  the  glass  sphere  and  the  experiment  with  the  knots.  To 
explain  the  connection  of  the  glass  sphere  experiment  with  the  fourth 


104  THE  FOURTH  DIMENSION. 

dimension,  we  must  first  conceive  of  two-dimensional  reasoning  be- 
ings, or,  let  us  say,  two-dimensional  worms,  living  and  moving  in  a 
plane.  For  a  creature  of  this  kind  it  will  be  self-evident  that  there 
are  no  other  paths  between  two  points  of  its  plane  than  such  as  lie 
within  the  plane.  It  must,  accordingly,  be  beyond  the  range  of 
conception  of  this  worm,  how  any  two-dimensional  object  which  lies 
within  a  circle  in  its  space  can  be  brought  to  any  other  position  in 
its  space  outside  the  circle  without  the  object  passing  through  the 
barriers  formed  by  the  circle's  circumference.  But  if  this  worm 
could  procure  the  services  of  a  three-dimensional  being,  the  trans- 
portation of  the  object  from  a  position  within  the  circle  to  any  posi- 
tion outside  it  could  be  effected  by  the  three-dimensional  being  sim- 
ply taking  the  object  out  <?/the  plane  and  placing  it  at  the  desired 
point.  This  object,  therefore,  would,  in  an  inexplicable  manner, 
suddenly  disappear  before  the  eyes  of  the  worms  who  were  assem- 
bled as  spectators,  and  after  the  lapse  of  an  interval  of  time  would 
again  appear  outside  the  circle  without  having  passed  at  any  point 
through  the  circle's  circumference.  If  now  we  add  another  dimen- 
sion, we  shall  derive  from  this  trick,  which  is  wholly  removed  from 
the  sense-perception  of  the  flattened  worms,  the  following  experi- 
ment, which  is  wholly  beyond  the  perception  of  us  human  beings. 
Inside  a  glass  sphere,  which  is  closed  all  around,  a  grain  of  corn  is 
placed  ;  the  problem  is  to  transport  the  corn  to  some  place  outside 
the  sphere  without  passing  through  the  glass  surface.  Now  we 
should  be  able  to  perform  this  trick  if  some  four-dimensional  being 
would  render  us  the  same  aid  that  we  before  rendered  the  two- 
dimensional  worm.  For  the  four-dimensional  being  could  take  the 
grain  of  corn  into  his  four-dimensional  space  and  then  replace  it  in 
our  space  in  the  desired  spot  outside  of  the  glass  sphere.  Slade 
performed  this  trick  before  Zollner.  Its  mere  performance  sufficed 
to  convince  this  latter  investigator  that  Slade  had  here  made  use  of 
a  four-dimensional  agent,  who,  in  respect  of  power  of  motion,  con- 
trolled his  four-dimensional  space  as  we  do  our  three-dimensional 
space.  It  never  occurred  to  Zollner  that  this  experiment  was  the 
cleverly  executed  trick  of  a  prestidigitateur,  or,  as  it  would  at  once 
occur  to  us,  that  the  whole  thing  was  a  sensor}-  illusion.      The  fact 


THE  FOURTH  DIMENSION.  IO5 

that  we  cannot  explain  a  trick  easily  and  naturally  does  not  irrevo- 
cably prove  that  it  is  accomplished  by  other  means  than  those 
which  the  world  of  matter  presents. 

Still  better  known  than  this  last  performance  is  Slade's  experi- 
ments with  knots.  To  explain  this  in  connection  with  the  fourth 
dimension,  we  must  resort  again  to  the  plane  and  the  flat  worm  in- 
habiting it.  To  two  parallel  lines  in  a  plane  let  the  two  ends  of  a 
third  line,  which  has  a  double  point,  that  is,  intersects  itself  once, 
be  attached.  Our  flat  worm  would  not  be  able  to  untie  the  loop 
formed  by  the  doubled  third  line,  which  we  will  call  a  string,  be- 
cause it  cannot  execute  motions  in  three  dimensions.  If,  therefore, 
a  two-dimensional  prestidigitateur  should  appear  and  accomplish 
the  trick  of  untying  this  loop  without  removing  the  two  ends  of  the 
string  from  the  parallel  lines,  he  will  have  accomplished  for  our  flat 
worm  a  supersensuous  experiment.  A  human  being  engaged  in  the 
service  of  the  prestidigitateur  could  execute  for  him  the  experiment 
by  simply  lifting  the  string  a  little  out  of  the  plane,  pulling  it  taut, 
and  placing  it  back  again.  This  suggests  the  following  analogous 
experiment  for  three-dimensional  beings.  The  two  ends  of  a  string 
in  which  a  common  knot  has  been  made  are  sealed  to  the  opposite 
walls  of  a  room.  The  problem  is  to  untie  this  knot  without  break- 
ing the  seals  at  the  two  ends  of  the  string.  Everybody  knows  that 
this  problem  is  not  soluble,  but  it  may  be  calculated  mathemati- 
cally that  the  knot  in  the  string  can  be  untied  as  easily  by  motions 
in  a  fourth  dimension  of  space  as  in  the  experiment  above  described 
the  knot  in  the  two-dimensional  string  was  untied  by  a  three-dimen- 
sional motion.  Now  as  Slade  untied  the  knot  before  Zollner's  eyes 
without  apparently  making  any  use  of  the  ends  fastened  in  the 
walls,  Zollner  was  still  more  firmly  confirmed  in  the  view  that  Slade 
had  power  over  spirits  who  performed  the  experiments  for  him. 

Still  more  far-reaching  is  the  theory  of  Carl  du  Prel  concerning 
the  relations  of  the  material  and  the  four-dimensional  world.  (Com- 
pare his  numerous  essays  in  the  spiritualistic  magazine  Sphinx.) 
Just  as  the  shadows  of  three-dimensional  objects  cast  on  a  wall  are 
controlled  in  their  movements  by  the  things  whose  projections  they 
are,  in  the  same  way  it  is  claimed  does  there  exist  back  of.  every- 


Io6  THE  FOURTH  DIMENSION. 

thing  of  this  sense-perceptible  world  a  real  transcendental  and  four 
dimensional  "  thing-in-itself "  whose  projection  in  the  space  of  ex- 
perience is  what  we  falsely  regard  as  the  independent  thing.  Thus 
every  man  besides  existing  in  his  terrestrial  self  also  exists  in  a 
spiritual  or  astral  self  which  constantly  accompanies  him  in  his 
walks  through  life  and  whose  existence  is  especially  proclaimed  in 
states  of  profound  sleep,  of  somnambulism,  and  in  the  conditions 
of  mediums.  In  this  way  Du  Prel  explains  the  wonderful  feats  of 
somnambulists,  and  the  aerial  journeys  of  sorcerers  and  witches. 
Whereas,  ordinarily  the  separation  of  the  material  body  from  the 
astral  body  is  only  effected  at  the  moment  of  death  ;  in  the  case  of 
somnambulists  this  separation  may  take  place  at  any  time,  or,  as  Du 
Prel  says,  "the  threshold  of  feeling  may  be  permanently  displaced." 

In  view  of  the  natural  relations  of  such  theories  to  the  dogmas 
of  Christianity  it  is  explainable  that  theologians  also  have  raised 
their  voices  for  or  against  spiritualism.  While  the  Protestatit  Church 
Times  beheld  in  the  "repulsive  thaumaturgic  performances  which 
these  coryphaei  of  modern  science  offer,  a  lack  of  comprehension  of 
real  philosoph}', "  the  magazine  The  Proof  of  Faith  expresses  its  de- 
light at  the  discovery  of  spiritualism  in  the  following  manner : 
"Every  Christian  will  surely  rejoice  at  the  deep  and  perhaps  mor- 
tal wound  which  these  new  discoveries  have  in  all  probability  ad- 
ministered to  modern  materialism." 

We  shall  pass  by  the  childish  opinion  that  the  Bible  also  speaks 
of  four  dimensions,  as  both  in  Job  (xi.  8-9)  and  in  the  Epistle  to 
the  Ephesians  (iii.  18)  only  breadth,  length,  depth,  and  height, 
that  is,  four  directions  of  extension,  are  mentioned.  Yet  we  will 
still  add,  as  Cranz  has  done,  the  reflections  which  Zollner,  as  the 
most  prominent  representative  of  modern  spiritualism,  has  put  for- 
ward respecting  its  relations  to  the  doctrines  of  Christianity  (  Wis- 
sensch.  Abhandl.,  Vol.  III).  By  the  foundation  of  transcendental 
physics  on  the  basis  of  spiritualistic  phenomena,  the  "new  light" 
has  arisen  which  is  spoken  of  in  the  New  Testament.  The  rending  of 
the  veil  of  the  Temple  on  the  crucifixion  of  Christ,  the  resurrection, 
the  ascension,  the  transfiguration,  the  speaking  with  many  tongues 
on  the  giving  out  of  the  Holy  Ghost,  all  these  are  in  Zollner's  view 


THE   FOURTH  DIMENSION.  ID/ 

spiritualistic  phenomena.  Similarly,  he  sees  a  reference  to  the 
four-dimensional  world  of  spirits  in  all  those  sayings  of  Christ  in 
which  the  latter  speaks  to  his  Apostles  of  the  impossibility  of  their 
having  any  image  or  notion  of  the  place  to  which  when  he  dis- 
appeared he  would  go  and  whence  he  would  return.  (Gospel  of 
St.  John,  xii.  33,  36;   xiv.  2,  3,  28;  xvi.  5,  13.) 

Ulrici,  however,  goes  farthest  in  the  mingling  of   spiritualistic 
and  Christian  beliefs ;  for  he  sees  in  the  doctrine  of  spiritualism  a 
means  of  strengthening  belief  in  a  supreme  moral  world-order  and 
in  the  immortality  of  the  soul.      In  answer  to  Ulrici's  tract  "Spirit- 
ualism  So-called,  a  Question   of  Science"  (1889)  Wundt  wrote  an 
annihilating  reply  bearing  the   title  "Spiritualism,    a  Question  of 
Science  So-called."     Wundt  criticises  the  future  condition  of  our 
soul  according  to  spiritualistic  hypotheses  in  the  following  sarcastic 
yet  pertinent  words,  which  Cranz  also  quotes:  "(i)  Physically,  the 
"souls  of  the  dead  come  into  the  thraldom  of  certain  living  beings 
"who  are  called  mediums.     These  mediums  are,  for  the  present  at 
"least,  a  not  widely  diffused  class,  and  they  appear  to  be  almost 
"exclusively  Americans.     At  the  command  of  these  mediums,  de- 
"p-arted  souls  perform  mechanical  feats  which  possess  throughout 
"the  character  of  absolute  aimlessness.      They  rap,  they  lift  tables 
"and  chairs,  they  move  beds,  they  play  on  the  harmonica,  and  do 
"other  similar  things.      (2)  Intellectually,   the  souls  of  the  dead 
"enter  a  condition  which,  if  we  are  to  judge  from  the  productions 
"which  they  deposit  on  the  slates  of  the  mediums,  must  be  termed 
"a  very  lamentable  one.      These  slate-writings  belong  throughout 
"in  the  category  of  imbecility  ;  they  are  totally  bereft  of  any  con- 
"  tents.    (3)  The  most  favored,  apparently,  is  the  moral  condition  of 
"the  soul.     According  to  the  testimony  which  we  have,  its  charac- 
"ter  cannot  be  said  to  be  anything  else  than  that  of  harmlessness. 
"From  brutal  performances,  such,  for  instance,  as  the  destruction 
"of  bed-canopies,  the  spirits  most  politely  refrain."     Wundt  then 
laments  the  demoralising  effect  which  spiritualism  exercises  on  peo- 
ple who  have  hitherto  devoted  their  powers  to  some  serious  pursuit 
or  even  to  the  service  of  science.      In  fact  it  is  a  presumptuous  and 
flagrant  procedure  to  forsake  the  path  of  exact  research,  which 


OF  THE  '   ^^ 


I08  THE  FOURTH  DIMENSION. 

slow  as  it  is,  yet  always  leads  to  a  sure  extension  of  knowledge,  in 
the  hope  that  some  aimless,  foolhardy  venture  into  the  realm  of 
uncertainty  will  carry  us  farther  than  the  path  of  slow  toil,  and  that 
we  can  ever  thus  easily  lift  the  veil  which  hides  from  man  the  prob- 
lems of  the  world  that  are  yet  unsolved. 

* 
*  * 

Now  that  we  have  presented  the  opinions  of  others  respecting 
the  existence  of  a  four  dimensional  world  of  spirits,  the  author  would 
like  to  develop  one  or  two  ideas  of  his  own  on  the  subject.  In  the 
preceding  section  it  was  stated  that  everything  that  we  perceive  by 
our  senses  is  three-dimensional  and  that  everything  that  possesses 
four  or  more  dimensions  can  only  be  regarded  as  abstractions  or  fic- 
tions which  our  reason  forms  in  its  constant  efforts  after  an  exten- 
sion and  generalisation  of  knov/ledge.  To  speak  of  two-dimensional 
matter  is  as  self-contradictory  as  the  notion  of  four-dimensional 
matter.  But  a  two  or  a  four-dimensional  world  might  exist  in  some 
other  manner  than  a  material  manner,  and  for  all  we  know  in  one 
which  to  us  does  not  admit  of  representation.  But  in  such  a  case, 
if  it  were  without  the  power  of  affecting  the  material  world,  we  should 
never  be  able  to  acquire  any  knowledge  concerning  its  existence, 
and  it  would  be  totally  indifferent  to  the  people  of  the  three-dimen- 
sional world,  whether  such  a  world  existed  or  not.  Just  as  an  artist 
during  his  lifetime  produces  a  number  of  different  works  of  art,  so 
also  the  Creator  might  have  created  a  number  of  different-dimen- 
sioned worlds  which  in  no  wise  interfere  with  one  another.  In  such 
a  case,  any  one  world  would  not  be  able  to  know  anything  of  any 
other,  and  we  must  consequently  regard  the  question  whether  a 
four-dimensional  world  exists  which  is  incapable  of  affecting  ours, 
as  insoluble.  We  have  only  to  examine,  therefore,  the  question 
whether  the  phenomenal  world  perhaps  is  a  single  individual  in  a 
great  layer  of  worlds  of  which  every  successive  one  has  one  more 
dimension  than  the  preceding  and  which  are  so  connected  with  one 
another  that  each  successive  world  contains  and  includes  the  pre- 
ceding world,  and,  therefore,  can  produce  effects  in  it.  For  our 
reason,  which  draws  its  inferences  from  the  phenomena  of  this  world, 
tells  us,  that  if  outside  the  three-dimensional  world  there  exists  a 


THE  FOURTH  DIMENSION.  lOg 

second  four-dimensional  world,  containing  ours,  there  is  no  reason 
why  worlds  of  more  or  less  dimensions  should  not,  with  the  same 
right,  also  exist.  But  if  now,  as  Zollner  and  his  adherents  main- 
tain, four-dimensional  spirits  exist  which  can  act  by  the  mere  efforts 
of  their  own  wills  on  our  world,  there  is  surely  no  reason  why  we 
three-dimensional  beings  should  not  also  be  able  to  produce  effects 
on  some  two-dimensional  animated  world.  Whether  we  have,  gen- 
erally, any  such  influence  we  do  not  know,  but  we  certainly  do  know 
that  we  do  not  act  purposely  and  consciously  on  a  two-dimensional 
world.  If,  therefore,  Zollner  were  right,  the  plan  of  creation  would 
possess  the  wonderful  feature  that  four-dimensional  beings  are  cap- 
able of  arbitrarily  affecting  the  three-dimensional  world,  but  that 
three-dimensional  beings  have  no  right  in  their  turn  consciously  to 
affect  a  two-dimensional  world. 

The  supporters  of  Zollner's  hypothesis  will  perhaps  reply  to 
the  objection  just  made,  that  the  plan  of  creation  might,  after  all, 
possibly  possess  this  wonderful  peculiarity,  that  we  human  beings 
perhaps,  in  some  higher  condition  of  culture,  will  be  able  to  act  con- 
sciously on  two-dimensional  worlds,  and  that  at  any  rate  it  is  simply 
an  inference  by  analogy  to  conclude  from  the  non-existence  of  a  rela- 
tion between  three  and  two  dimensions  that  the  same  relation  is  also 
wanting  in  the  case  of  four  and  three  dimensions.  As  a  matter  of  fact, 
the  objection  above  made  is  not  intended  to  refute  Zollner's  hypoth- 
esis, but  only  to  stamp  it  as  very  improbable.  But  despite  this  im- 
probability Zollner  would  still  be  right  if  the  phenomena  of  the  ma- 
terial world  actually  made  his  hypothesis  necessary.  That,  however, 
is  by  no  means  the  case.  Although  most  of  the  phenomena  to  which 
the  spiritualists  appeal  are  probably  founded  on  sense-illusions, 
humbug,  and  self-deception,  it  cannot  be  denied  that  there  possibly 
do  exist  phenomena  which  cannot  be  brought  into  harmony  with 
the  natural  laws  now  known.  There  always  have  been  mysterious 
phenomena,  and  there  always  will  be.  Yet,  as  we  have  often  seen 
that  the  progress  of  science  has  again  and  again  revealed  as  natural 
what  former  generations  held  to  be  supernatural,  it  is  certainly 
wholly  wrong  to  bring  in  for  the  explanation  of  phenomena  which 
now  seem   mysterious  an  hypothesis  like  that  of  Zollner,  by  which 


no  THE  FOURTH  DIMENSION. 

everything  in  the  world  can  be  explained.  If  we  adopt  a  point  of 
view  which  regards  it  as  natural  for  spirits  arbitrarily  to  interfere  in 
the  workings  of  the  world,  all  scientific  investigation  will  cease,  for 
we  could  never  more  trust  or  rely  upon  any  chemical  or  physical  ex- 
periment, or  any  botanical  or  zoological  culture.  If  the  spirits  are 
the  authors  of  the  phenomena  that  are  mysterious  to  us,  why  should 
they  also  not  have  control  of  the  phenomena  which  are  not  myste- 
rious? The  existence  of  mysterious  phenomena  justifies  in  no  man- 
ner or  form  the  assumption  that  spirits  exist  which  produce  them. 
Would  it  not  be  much  simpler,  if  we  must  have  supernatural  in- 
fluences, to  adopt  the  naive  religious  point  of  view,  according  to 
which  everything  that  happens  is  traceable  to  the  direct,  actual,  and 
personal  interference  of  a  single  being  which  we  call  God?  Things 
which  formerly  stood  beyond  the  sphere  of  our  knowledge  and  were 
regarded  as  marvellous  events,  as  a  storm,  for  example,  now  stand 
in  the  most  intimate  connection  with  known  natural  laws.  Things 
that  formerly  were  mysterious  are  so  no  longer.  If  one  hundred  and 
fifty  years  ago  some  scientists  were  in  the  possession  of  our  present 
knowledge  of  inductional  electricity  and  had  connected  Paris  and 
Berlin  with  a  wire  by  whose  aid  one  could  clearly  interpret  in 
Berlin  what  another  person  had  at  that  very  moment  said  in  Paris, 
people  would  have  regarded  this  phenomenon  as  supernatural  and 
assumed  that  the  originator  of  this  long-distance  speaking  was  in 
league  with  spirits. 

We  recognise,  thus,  that  the  things  which  are  termed  super- 
natural depend  to  a  great  extent  on  the  stage  of  culture  wliich  hu- 
manity has  reached.  Things  which  now  appear  to  us  mysterious, 
may,  in  a  very  few  decades,  be  recognised  as  quite  natural.  This 
knowledge,  however,  is  not  to  be  obtained  by  the  lazy  assumption 
of  bands  of  spirits  as  the  authors  of  mysterious  phenomena,  but  by 
performing  what  in  physics  and  chemistry  is  called  experiment. 
But  the  first  and  essential  condition  of  all  scientific  experimenting 
is  that  the  experimenter  shall  1  e  absolutely  master  of  the  conditions 
under  which  the  experiment  is  or  is  not  to  succeed.  Now,  this  cri- 
terion of  scientific  experimenting  is  totally  lacking  in  all  spiritualistic 
experiments.    We  can  never  assign  in  their  case  the  conditions  un- 


THE  FOURTH  DIMENSION.  Ill 

der  which  they  will  or  will  not  succeed.  When  all  the  preparations 
in  a  spiritualistic  stance  have  been  properly  made,  but  nothing  takes 
place,  the  beautiful  excuse  is  always  forthcoming  that  the  "spirits 
were  not  willing,"  that  there  were  "too  many  incredulous  persons 
present,"  and  so  forth.  Fortunately,  in  physical  experiments  these 
pretexts  are  not  necessary.  By  the  path  of  experiment,  and  not  by 
that  of  transcendental  speculation,  physics  has  thus  made  incredible 
progress  and  has  piled  new  knowledge  strata  on  strata  upon  the 
old.  Accordingly,  the  prospect  is  left  that  the  mysteries  which  the 
conditions  and  properties  of  the  human  soul  still  present  can  be 
solved  more  and  more  by  the  methods  of  scientific  experiment.  To 
this  end,  however,  it  is  especially  necessary  that  the  physio-psycho- 
logical experiments  in  question  should  only  be  performed  by  men 
who  possess  the  critical  eye  of  inquiry,  who  are  free  from  the  dan- 
gers of  self-illusion,  and  who  are  competent  to  keep  apart  from  their 
experiments  all  superstition  and  deception.  The  history  of  natural 
science  clearly  teaches  that  it  is  only  by  this  road  that  man  can  ar- 
rive at  certain  and  well-established  knowledge.  If,  therefore,  there 
really  is  behind  such  phenomena  as  mind-reading,  telepathy,  and 
similar  psychical  phenomena,  something  besides  humbug  and  self- 
illusion,  what  we  have  to  do  is  to  study  privately  and  carefully  by 
serious  experiments  the  success  or  non-success  of  such  phenomena, 
and  not  allow  ourselves  to  be  influenced  by  the  public  and  dramatic 
performances  of  psychical  artists,  like  Cumberland  and  his  ilk. 

The  high  eminence  on  which  the  knowledge  and  civilisation  of 
humanity  now  stands  was  not  reached  by  the  thoughtless  employ- 
ment of  fanciful  ideas,  nor  by  recourse  to  four-dimensional  worlds, 
but  by  hard,  serious  labor,  and  slow,  unceasing  research.  Let  all  men 
of  science,  therefore,  band  themselves  together  and  oppose  a  solid 
front  to  methods  that  explain  everything  that  is  now  mysterious  to 
us  by  the  interference  of  independent  spirits.  For  these  methods, 
owing  to  the  fact  that  they  can  explain  everything,  explain  nothing, 
and  thus  oppose  dangerous  obstacles  to  the  progress  of  real  research, 
to  which  we  owe  the  beautiful  temple  of  modern  knowledge. 


THE  SQUARING  OF  THE  CIRCLE. 

AN  HISTORICAL   SKETCH  OF  THE  PROBLEM  FROM  THE  RE- 
MOTEST TIMES. 
I. 
UNIVERSAL  INTEREST  IN  THE  PROBLEM. 

FOR  two  and  a  half  thousand  years,  both  competent  and  incom- 
petent minds  have  striven  in  vain  to  solve  the  problem  known 
as  the  squaring  of  the  circle.  ^Jow  that  geometers  have  at  last  suc- 
ceeded in  giving  a  rigorous  demonstration  of  the  impossibility  of 
solving  the  problem  with  straight  edge  and  compasses,  it  seems 
fitting  and  opportune  to  cast  a  glance  into  the  nature  and  history 
of  this  very  ancient  problem.  And  this  will  be  found  the  more  jus- 
tifiable in  view  of  the  fact  that  the  squaring  of  the  circle,  at  least 
in  name,  is  very  widely  known  outside  of  the  narrow  circle  of  pro- 
fessional mathematicians. 

The  Proceedings  of  the  French  Academy  for  the  j'ear  1775  con- 
tain at  page  61  a  resolution  of  the  Academy  not  to  examine  from 
that  time  on,  any  so-called  solutions  of  the  quadrature  of  the  circle. 
The  Academy  was  driven  to  this  determination  by  the  overwhelm- 
ing multitude  of  professed  solutions  of  the  famous  problem  which 
were  sent  to  it  every  month  in  the  year, — solutions  which  of  course 
were  an  invariable  attestation  of  the  ignorance  and  self-conceit  of 
their  authors,  but  which  suffered  collectively  from  the  very  impor- 
tant drawback  in  mathematics  of  being  wrong.  Since  that  time  all 
professed  solutions  of  the  problem  received  by  the  Academy  find  a 
sure  and  permanent  resting-place  in  the  waste-basket,  and  remain 
unanswered  for  all  time.      The  circle-squarer,  however,  sees  in  this 


THE  SQUARING  OF  THE  CIRCLE.  113 

high-handed  manner  of  rejection  only  the  envy  of  the  great  and 
powerful  at  his  grand  intellectual  disco^i^ery.  He  is  determined  to 
secure  recognition,  and  appeals  therefore  to  the  public.  The  news- 
papers must  obtain  for  him  the  appreciation  that  scientific  societies 
have  denied.  And  every  year>  the  old  mathematical  sea-serpent 
more  than  once  disports  itsel*  in  the  columns  of  our  newspapers  in 
the  shape  of  an  announcement  that  Mr.  N.  N.,  of  P.  P.,  has  at  last 
solved  the  problem  of  the  quadrature  of  the  circle. 

But  what  manner  of  people  are  these  circle-squarers,  when  ex- 
amined by  the  light^i:-  Almost  always  they  will  be  found  to  be  im- 
perfectly educated  -persons,  whose  mathematical  knowledge  does 
not  exceed  that  of  a  modern  high-school  student.  It  is  seldom  that 
they  know  accurately  what  the  requirements  of  the  problem  are  and 
what  its  nature-  'tiiey  are  totally  ignorant  of  the  two  and  a  half 
thousand  yeSits'  history  of  the  problem  ;  and  they  have  no  idea 
whatever  of  .the  ■important  investigations  which  have  been  made 
with  regard'to  it 'by  great  and  real  mathematicians  in  every  century 
down  to  our  own  time.* 

Yet  great  as  is  the  quantum  of  ignorance  that  circle-squarers 
intermix  witii^ their  intellectual  products,  the  lavish  supply  of  con- 
ceit and  egotism  with  which  they  season  their  performances  is  still 
greater.;  I  have  not  far  to  go  to  furnish  a  verification  of  this.  A 
book  printed  in  Hamburg  in  the  year  1840  lies  before  me,  in  which 
the  author  thanks  Almighty  God  at  every  second  page  that  He  has 
selecteid  him  and  no  one  else  to  solve  the  "  problem  phenomenal" 
of  ma;thematics,  "so  long  sought  for,  so  fervently  desired,  and  at- 
tempted'by  millions."  After  this  modest  author  has  proclaimed 
himiSelf  the  unmasker  of  Archimedes's  deceit,  he  says  :  "And  thus 
it  hath  pleased  our  mother  Nature  to  withhold  this  precious  math- 
emiitical  jewel  from  the  eye  of  human  investigation,  until  she 
tho4ight  it  fitting  to  reveal  truth  to  simplicity." 

't  This  will  suffice  to  show  the  great  fatuity  of  the  author.      But 
it.dSDes  not  suffice  to  prove  his  ignorance.      He  has  no  conception 


"  '  *  I'or  the  full  psychogeny  and  psychiatry  of  the  circle-squarer  see  A.  De  Mor- 
gan, /I  Budget  of  Paradoxes  (London,  1872).  —  Tr. 


w 


114  THE   SQUARING  OF  THE  CIRCLE. 

I  ot  mathematical  demonstration  ;  he  takes  it  for  granted  that  things 
are  so  because  they  seem  so  to  him.  Errors  of  logic,  also,  abound 
in  his  book.  But,  minor  fallacies  apart,  wherein  does  the  real  error 
of  this  "  unmasker  "  of  Archimedes  consist?  It  requires  consider- 
able labor  to  extricate  the  kernel  of  the  demonstration  from  the 
turgid  language  and  bombastic  style  in  which  the  author  has  buried 
his  conclusions.  But  it  is  this.  The  author  inscribes  a  square  in  a 
circle,  circumscribes  another  about  it,  then  points  out  that  the  inside 
square  is  made  up  of  four  congruent  triangles,  whereas  the  circum- 
scribed square  is  made  up  of  eight  such  triangles  ;  from  which  fact, 
seeing  that  the  circle  is  larger  than  the  one  square  and  smaller  than 
the  other,  he  draws  the  bold  conclusion  that  the. circle  is  equal  in 
■^  area  to  six  such  triangles.  It  is  hardly  conceivable  that  a  rational 
being  could  infer  that  something  which  is  greater.vthan  4  and  less 

;  than  8  must  necessarily  be  6.  But  with  a  man  that  attempts  the 
squaring  ol  the  circle  this  kind  of  ratiocination  /s  piassible. 

It  is  the  same  with  all  the  other  attempted  solutions  of  the 
problem  ;  in  all  of  them  either  logical  fallacies  or  violabjons  of  ele- 
mentary arithmetical  or  geometrical  truths  can  be  pointed  out. 
Only  they  are  not  always  of  such  a  trivial  nature  as  in  the  book 
just  mentioned.  ^ 

Let  us  now  inquire  into  the  origin  of  this  propensity  •which 

leads  people  to  occupy  themselves  with  the  quadrature  of  th«  circle. 

Attention  must  first  be  called  to  the  antiquity  of  the  pr'oblem. 

I      A  quadrature  was  attempted  in  Egypt  500  years  before  the.  ^xodus 

I  of  the  Israelites.  Among  the  Greeks  the  problem  never  cealsed  to 
play  a  part  that  greatly  influenced  the  progress  of  mathem';atics. 
And  in  the  middle  ages  also  the  squaring  of  the  circle  sporadically 
appears  as  the  philosopher's  stone  of  mathematics.      The  problem 

j  has  thus  never  ceased  to  be  dealt  with  and  considered.  But  it 
is  not  by  the  antiquity  of  the  problem  that  circle-squarers  are^: en- 
ticed, but  by  the  allurement  which  everything  exerts  that  is  cal- 
culated to  raise  the  individual  above  the  mass  of  ordinary  hunian- 
it}',  and  to  bind  about  his  temples  the  laurel  crown  of  celebrity. 
Ambition  spurred  men  on  in  ancient  Greece  and  still  spurs  them 
on   in   modern    times   to   crack    this    primeval    mathematical'  nut. 


THE   SQUARING  OF  THE  CIRCLE.  115 

Whether  they  are  competent  thereto  is  a  secondary  consideration. 
They  look  upon  the  squaring  of  the  circle  as  the  grand  prize  of  a 
lottery  that  can  just  as  well  fall  to  their  lot  as  that  of  any  other 
man.      They  do  not  remember  that — 

"Toil  before  honor  is  placed  by  sagacious  decrees  of  Immortals." 

and  that  it  requires  years  of  consecutive  study  to  gain  possession  of 
the  mathematical  weapons  that  are  indispensably  necessary  to  at- 
tack the  problem,  but  which  even  in  the  hands  of  the  most  distin- 
guished mathematical  strategists  did  not  suffice  to  take  the  strong- 
hold. 

But  why  is  it,  we  must  further  ask,  that  it  happens  to  be  the 
squaring  of  the  circle  and  not  some  other  unsolved  mathematical 
problem  upon  which  the  efforts  of  people  are  bestowed  who  have 
no  knowledge  of  mathematics  yet  busy  themselves  with  mathemati' 
cal  questions?  The  question  is  answered  by  the  fact  that  the  squar- 
ing of  the  circle  is  about  the  only  mathematical  problem  that  is 
known  to  the  unprofessional  world, — at  least  by  name.  Even  among 
the  Greeks  the  problem  was  very  widely  known  outside  of  mathe- 
matical circles.  In  the  eyes  of  the  Grecian  layman,  as  at  present 
among  many  of  his  modern  brethren,  occupation  with  this  problem 
was  regarded  as  the  most  important  and  essential  business  of 
mathematicians.  In  fact,  they  had  a  special  word  to  designate  this 
species  of  activity,  namely,  Terpaywi't'^eiv,  which  means  to  busy  one's 
self  with  the  quadrature.  In  modern  times,  also,  every  educated 
person,  though  he  be  not  a  mathematician,  knows  the  problem  by 
name,  and  knows  that  it  is  insoluble,  or  at  least,  that  despite  the 
efforts  of  the  most  famous  mathematicians  it  has  not  yet  been 
solved.  For  this  reason  the  phrase  "to  square  the  circle,"  is  now 
generally  used  in  the  sense  of  attempting  the  impossible. 

But  in  addition  to  the  antiquity  of  the  problem,  and  the  fact 
also  that  it  is  known  to  the  lay  world,  there  is  an  important  third 
factor  that  induces  people  to  occupy  themselves  with  it.  This  is  the 
report  that  has  been  current  for  more  than  a  century  now,  that  the 
Academies,  the  Queen  of  England,  or  some  other  influential  person 
has  offered  a  large  prize  to  be  given  to  the  one  that  first  solves  the 


Il6  THE   SQUARING  OF  THE  CIRCLE. 

problem.  As  a  matter  of  fact,  the  hope  of  obtaining  this  large 
prize  of  money  is  with  many  circle-squarers  the  principal  incitement 
to  action.  And  the  author  of  the  book  above  referred  to  begs  his 
readers  to  lend  him  their  assistance  in  obtaining  the  prizes  offered. 

Although  the  opinion  is  widely  current  in  the  unprofessional 
world,  that  professional  mathematicians  are  still  busied  with  the 
solution  of  the  problem,  this  is  by  no  means  the  case.  On  the  con- 
trary, for  some  two  hundred  years,  the  endeavors  of  many  great 
mathematicians  have  been  directed  solely  towards  demonstrating 
with  exactness  that  the  problem  is  insoluble.  It  is,  as  a  rule, — 
and  naturally, — more  difficult  to  prove  that  a  thing  is  impossible 
than  to  prove  that  it  is  possible.  And  thus  it  has  happened,  that 
up  to  within  a  few  years  ago,  despite  the  employment  of  the  most 
varied  and  the  most  comprehensive  methods  of  modern  mathemat- 
ics, no  one  succeeded  in  supplying  the  wished-for  demonstration  of 
the  problem's  impossibility.  At  last.  Professor  Lindemann,  of 
Konigsberg,  in  June,  1882,  succeeded  in  furnishing  a  demonstra- 
tion,— and  the  first  demonstration, — that  it  is  impossible  by  em- 
ploying only  straight  edge  and  compasses  to  construct  a  squaer 
that  is  mathematically  exactly  equal  in  area  to  a  given  circle.  The 
demonstration,  naturally,  was  not  effected  with  the  help  of  the  old 
elementary  methods ;  for  if  it  were,  it  would  have  been  accom- 
plished centuries  ago ;  but  methods  were  requisite  that  were  first 
furnished  by  the  theory  of  definite  integrals  and  departments  of 
higher  algebra  developed  in  the  last  few  decades;  in  other  words, 
it  required  the  direct  and  indirect  preparatory  labor  of  many  cen- 
turies to  make  finally  possible  a  demonstration  of  the  insolubility 
of  this  historic  problem. 

Of  course,  this  demonstration  will  have  no  more  effect  than 
the  resolution  of  the  Paris  Academy  of  1775,  in  causing  the  fecund 
race  of  circle-squarers  to  vanish  from  the  face  of  the  earth.  In  the 
future  as  in  the  past,  there  will  be  people  who  know  nothing  of 
this  demonstration  and  will  not  care  to  know  anything,  and  who 
believe  that  they  cannot  help  succeeding  in  a  matter  in  which  oth- 
ers have  failed,  and  that  just  they  have  been  appointed  by  Provi- 
dence to  solve  the  famous  puzzle.      But  unfortunately  the  inerad- 


THE  SQUARING  OF  THE  CIRCLE.  I17 

icable  mania  for  solving  the  quadrature  of  the  circle  has  also  its 
serious  side.  Circle-squarers  are  not  always  so  self-satisfied  as 
the  author  of  the  book  above  mentioned.  They  often  see,  or  at 
least  divine,  the  insuperable  difficulties  that  tower  up  before  them, 
and  the  conflict  between  their  aspirations  and  their  performances, 
the  consciousness  that  the  problem  they  long  to  solve  they  are  un- 
able to  solve,  darkens  their  soul  and,  lost  to  the  world,  they  be- 
come interesting  subjects  for  the  science  of  psychiatry. 

IT. 
NATURE  OF  THE  PROBLEM. 

It  is  easy  to  determine  the  length  of  the  radius  of  a  circle,  or 
the  length  of  its  diameter,  which  must  be  double  that  of  the  radius; 
and  the  question  next  arises,  what  is  the  number  that  tells  how  many 
times  larger  the  circumference  of  the  circle,  that  is  the  length  of  the 
•circular  line,  is  than  its  radius  or  its  diameter.  From  the  fact  that 
all  circles  have  the  same  shape  it  follows  that  this  proportion  will 
be  the  same  for  all  circles  both  large  and  small.  Now,  since  the 
time  of  Archimedes,  all  civilised  nations  that  have  cultivated  math- 
ematics have  denoted  the  number  that  tells  how  many  times  larger 
the  circumference  of  a  circle  is  than  the  diameter  by  the  symbol  it, 
— the  Greek  initial  letter  of  the  word  periphery.*  To  compute  n, 
therefore,  means  to  calculate  how  many  times  larger  the  circumfer- 
ence of  a  circle  is  than  its  diameter.  This  calculation  is  called 
"the  numerical  rectification  of  the  circle." 

Next  to  the  calculation  of  the  circumference,  the  calculation  of 
the  superficial  contents  of  a  circle  by  means  of  its  radius  or  diam- 
eter is  perhaps  most  important;  that  is,  the  computation  of  how 
greAt  an,  area  that  part  of  a  plane  which  lies  within  a  circle  meas- 
ures. This  calculation  is  called  the  "numerical  quadrature."  It 
depends^  however,  upon  the  problem  of  numerical  rectification ; 
that  is,  upon  the  calculation  of  the  magnitude  of  n.  For  it  is  de- 
monstrated  in   elementary  geometry,   that  the   area  of  a  circle  is 

*The  Greek  symbol  tt  was  first  employed  by  W.  Jones  in  1706  and  did  not 
com;,  into  general  use  until  about  the  middle  of  the  eighteenth  century  through 
t'ha'' works  of  Euler. —  Trans. 


A 


Il8  THE  SQUARING  OF  THE  CIRCLE. 

equal  to  the  area  of  a  triangle  produced  by  drawing  in  the  circle  a 
radius,  erecting  at  the  extremity  of  the  same  a  tangent, — that  is,  in 
this  case,  a  perpendicular, — cutting,  off  upon  the  latter  the  length 
of  the  circumference,  measuring  from  the  extremity,  and  joining 
the  point  thus  obtained  with  the  centre  of  the  circle.  It  follows 
from  this  that  the  area  of  a  circle  is  as  many  times  larger  than  the 
square  upon  its  radius  as  the  number  n  amounts  to. 

The  numerical  rectification  and  numerical  quadrature  of  the 
circle  based  upon  the  computation  of  the  number  n  are  to  be  clearly 
distinguished  from  problems  that  require  a  straight  line  equal  in 
length  to  the  circumference  of  a  circle,  or  a  square  equal  in  area  to 
a  circle,  to  be  constricctively  produced  from  its  radius- or  its  diameter; 
problems  which  might  properly  be  called  "constructive  rectifica- 
tion" or  "constructive  quadrature."  Approximately,  of  course,  by 
employing  an  approximate  value  for  n,  these  problems  are  easily 
solvable.  But  to  solve  a  problem  of  construction  in  geometry, 
means  to  solve  it  with  mathematical  exactitude.  If  the  value  n 
were  exactly  equal  to  the  ratio  of  two  whole  numbers  to  each 
other,  the  constructive  rectification  would  present  no  difficulties. 
For  example,  suppose  the  circumference  of  a  circle  were  exactly 
31  times  greater  than  its  diameter  ;  then  the  diameter  could  be  di- 
vided into  seven  equal  parts,  which  could  easily  be  done  by  the 
principles  of  planimetry  with  straight  edge  and  compasses;  then 
by  prolonging  to  the  amount  of  such  a  part  a  straight  line  exactly 
three  times  as  long  as  the  diameter,  we  should  obtain  a  straight 
line  exactly  equal  to  the  circumference  of  the  circle.  But  as  a  mat- 
ter of  fact, — and  this  has  actually  been  demonstrated, — there  do 
not  exist  two  whole  numbers,  be  they  ever  so  great,  that  exactl)^ 
represent  by  their  proportion  to  each  other  the  number  ?r.  Con 
sequently,  a  rectification  of  the  kind  just  described  does  not  attain 
the  object  desired. 

It  might  be  asked  here,  whether  from  the  demonstrated  fact 
that  the  number  n  is  not  equal  to  the  ratio  of  two  whole  numbers 
however  great,  it  does  not  imimediately  follow  that  it  is  impossible 
to  construct  a  straight  line  exactly  equal  in  length  to  the  circuin- 
ference  of  a  circle  ;   thus  demonstrating  at  once  the  impossibility  of 


THE  SQUARING  OF  THE  CIRCLE.  II9 

solving  the  problem.  This  question  is  to  be  answered  in  the  nega- 
tive. For  in  geometry  there  can  easily  exist  pairs  of  lines  of  which 
the  one  can  be  readily  constructed  from  the  other,  notwithstanding 
the  fact  that  no  two  whole  numbers  can  be  found  to  represent  the 
ratio  of  the  two  lines.  The  side  and  the  diagonal  of  a  square,  for 
instance,  are  so  constituted.  It  is  true  the  ratio  of  the  latter  two 
magnitudes  is  nearly  that  of  5  to  7.  But  this  proportion  is  not 
exact,  and  there  are  in  fact  no  two  numbers  that  represent  the  ratio 
exactly.  Nevertheless,  either  of  these  two  lines  can  be  readily  con- 
structed from  the  other  by  employing  only  straight  edge  and  com- 
passes. This  might  be  the  case,  too,  with  the  rectification  of  the 
circle  ;  and  consequently  from  the  impossibility  of  representing  n 
by  the  ratio  between  two  whole  numbers  the  impossibility  of  the 
problem  of  rectification  is  not  inferable. 

The  quadrature  of  the  circle  stands  and  falls  with  the  problem 
of  rectification.  This  rests  upon  the  truth  above  mentioned,  that 
a  circle  is  equal  in  area  to  a  right-angled  triangle,  in  which  one 
side  is  equal  to  the  radius  of  the  circle  and  the  other  to  the  circum- 
ference. Supposing,  accordingly,  that  the  circumference  of  the  circle 
had  been  rectified,  then  we  could  construct  this  triangle.  But  every 
triangle,  as  we  know  from  plane  geometry,  can,  with  the  help  of 
straight  edge  and  compasses  be  converted  into  a  square  exactly 
equal  to  it  in  area.  So  that,  supposing  the  rectification  of  the  cir- 
cumference of  a  circle  to  have  beenT  successfully  effected,  a  square 
could  be  constructed  that  would  be  exactly  equal  in  area  to  the 
circle.  ^  uJ-:''^}''i^7''Jh-''^''ftlr>  ■>•    /<"*;f-^ 

The  dependence  upon  one  another  of  the  three  problems  of  the 
computation  of  the  number  tt,  the  quadrature  of  the  circle,  and  its 
rectification,  thus  obliges  us,  in  dealing  with  the  history  of  the 
quadrature,  to  regard  investigations  with  respect  to  the  value  of  ;r  (tv' 

and  attempts  to  rectify  the  circle  as  of  equal  importance,  and  to  f,     ,     f\ 

consider  them  accordingly. 

We  have  used  repeatedly  in  the  course  of  this  discussion  the 
expression  "  to  construct  with  straight  edge  and  compasses."  It 
will  be  necessary  to  explain  what  is  meant  by  the  specification  of 
these  two  instruments.     When  to  a   requirement  in   geometry  to 


i.20  THE  SQUARING  OF  THE  CIRCLE. 

construct  a  figure  there  are  so  large  a  number  of  conditions  an- 
nexed that  the  construction  of  only  one  figure  or  a  limited  number 
of  figures  is  possible  in  accordance  with  those  conditions  ;  such  a 
full  and  stated  requirement  is  called  a  problem  of  construction,  or 
briefly  a  problem.  When  a  problem  of  this  kind  is  presented  for 
solution  it  is  necessary  to  reduce  it  to  simpler  problems,  already 
recognised  as  solvable  ;  and  since  these  latter  depend  in  their  turn 
upon  other,  still  simpler  problems,  we  are  finally  brought  back  to 
certain  fundamental  problems  upon  which  the  rest  are  based  but 
which  are  not  themselves  reducible  to  problems  less  simple.  These 
fundamental  problems  are,  so  to  speak,  the  lowermost  stones  of  the 
edifice  of  geometrical  construction.  The  question  next  arises  as  to 
what  problems  may  be  properly  regarded  as  fundamental ;  and  it 
has  been  found,  that  the  solution  of  a  great  part  of  the  problems 
that  arise  in  elementary  plane  geometry  rests  upon  the  solution  of 
only  five  original  problems.     They  are  : 

1.  The  construction  of  a  straight  line  that  shall  pass  through 
two  given  points. 

2.  The  construction  of  a  circle  the  centre  of  which  is  a  given 
point  and  the  radius  of  which  has  a  given  length. 

3.  The  determination  of  the  point  lying  coincidently  on  two 
given  straight  lines  prolonged  as  far  as  necessary, — in  case  such  a 
point  (point  of  intersection)  exists. 

4.  The  determination  of  the  two  points  that  lie  coincidently 
on  a  given  straight  line  and  a  given  circle, — in  case  such  common 
points  (points  of  intersection)  exist. 

5.  The  determination  of  the  two  points  that  lie  coincidently  on 
two  given  circles, — in  case  such  common  points  (points  of  inter- 
section) exist. 

For  the  solution  of  the  three  last  of  these  five  problems  the 
eye  alone  is  needed,  while  for  the  solution  of  the  first  two  prob- 
lems, besides  pencil,  ink,  chalk,  or  the  like,  additional  special  in- 
struments are  required  :  for  the  solution  of  the  first  problem  a 
straight  edge  or  ruler  is  most  generally  used,  and  for  the  solution 
of  the  second  a  pair  of  compasses.  But  it  must  be  remembered 
that  it  is  no  concern  of  geometry  what  mechanical  instruments  are 


THE  SQUARING  OF  THE  CIRCLE,  121 

employed  in  the  solution  of  the  five  problems  mentioned.  Geom- 
etry simply  limits  itself  to  the  presupposition  that  these  problems 
are  solvable,  and  regards  a  complicated  problem  as  solved  if,  upon 
a  specification  of  the  constructions  of  which  the  solution  consists, 
no  other  requirements  are  demanded  than  the  five  above  mentioned. 
Since,  accordingly,  geometry  does  not  itself  furnish  the  solution  of 
these  five  problems,  but  rather  exacts  them,  they  are  termed  J>ostu- 
lates."^  All  problems  of  plane  geometry  are  not  reducible  to  these 
five  problems  alone.  There  are  problems  that  can  be  solved  only 
by  assuming  other  problems  as  solvable  which  are  not  included  in 
the  five  given  ;  for  example,  the  construction  of  an  ellipse,  having 
given  its  centre  and  its  major  and  minor  axes.  Many  problems, 
however,  possess  the  property  of  being  solvable  with  the  assistance 
of  the  above-formulated  five  postulates  alone,  and  where  this  is  the 
case  they  are  said  to  be  "  constructible  with  straight  edge  and  com- 
passes," or  "elementarily"  constructible. 

After  these  general  remarks  upon  the  solvability  of  problems 
of  geometrical  construction,  which  an  understanding  of  the  history 
of  the  squaring  of  the  circle  makes  indispensable,  the  significance 
of  the  question  whether  the  quadrature  of  the  circle  is  or  is  not 
solvable,  that  is  elementarily  solvable,  will  become  intelligible. 
But  the  conception  of  elementary  solvability  only  gradually  took 
clear  form,  and  we  therefore  find  among  the  Greeks  as  well  as 
among  the  Arabs  endeavors,  successful  in  some  respects,  that  aimed 
at  solving  the  quadrature  of  the  circle  with  other  expedients  than 
the  five  postulates.  We  have  also  to  take  these  endeavors  into 
consideration,  and  especially  so  as  they,  no  less  than  the  unsuccess- 
ful efforts  at  elementary  solution,  have  upon  the  whole  advanced 
the  science  of  geometry,  and  contributed  much  to  the  clarification 
of  geometrical  ideas. 


*  Usually  geometers  mention  only  two  postulates  (Nos.  i  and  2).  But  since  to 
geometry  proper  it  is  indifferent  whether  only  the  eye,  or  additional  special  mechan- 
ical instruments  are  necessary,  the  author  has  regarded  it  more  correct  in  point  of 
method  to  assume  five  postulates. 


122  THE  SQUARING  OF  THE  CIRCLE. 


IIJ. 
THE  EGYPTIANS,   BABYLONIANS,    AND  GREEKS. 

In  the  oldest  mathematical  work  that  we  possess  we  find  a  rule 
telling  us  how  to  construct  a  square  which  is  equal  in  area  to  a 
given  circle.  This  celebrated  book,  the  Rhind  Papuans  of  the  Brit- 
ish Museum,  translated  and  explained  by  Eisenlohr  (Leipsic,  1877), 
was  written,  as  stated  in  the  work  itself,  in  the  thirtj^-third  3'ear  of 
the  reign  of  King  Ra-a-us,  by  a  scribe  of  that  monarch,  named 
Ahmes.  The  composition  of  the  work  falls  accordingly  in  the  period 
of  the  two  Hyksos  dynasties,  that  is,  in  the  period  between  2000 
and  1700  B.  C.  But  there  is  another  important  circumstance  to  be 
noted.  Ahmes  mentions  in  his  introduction  that  he  composed  his 
work  after  the  model  of  old  treatises,  written  in  the  time  of  King 
Raenmat ;  whence  it  appears  that  the  originals  of  the  mathematical 
expositions  of  Ahmes  are  half  a  thousand  years  older  still  than  the 
Rhind  Papyrus. 

The  rule  given  in  this  papyrus  for  obtaining  a  square  equal  to 
a  circle  specifies  that  the  diameter  of  the  circle  shall  be  shortened 
one-ninth  of  its  length  and  upon  the  shortened  line  thus  obtained 
a  square  erected.  Of  course,  the  area  of  a  square  of  this  construc- 
tion is  only  approximately  equal  to  the  area  of  the  circle.  An  idea 
may  be  obtained  of  the  degree  of  exactness  of  this  original,  primi- 
tive quadrature  by  remarking,  that  if  the  diameter  of  the  circle  in 
question  is  one  metre  in  length,  the  square  that  is  supposed  to  be 
equal  to  the  circle  is  a  little  less  than  half  a  square  decimetre  too 
large ;  an  approximation  not  so  accurate  as  that  made  by  Archi- 
medes, yet  much  more  correct  than  many  a  one  later  employed.  It 
is  not  known  how  Ahmes  or  his  predecessors  arrived  at  this  ap- 
proximate quadrature  ;  but  it  is  certain  that  it  was  handed  down  in 
Egypt  from  century  to  century,  and  in  late  Egyptian  times  it  ap- 
pears repeatedly. 

In  addition  to  the  effort  of  the  Egyptians,  we  also  find  in  pre- 
Grecian  antiquity  an  attempt  at  circle-computation  among  the  Ba- 
bylonians.     This  is  not  a  quadrature,  but  is  intended  as  a  rectifica- 


THE  SQUARING  OF  THE  CIRCLE.  I23 

tion  of  the  circumference.  The  Babylonian  mathematicians  had 
discovered,  that  if  the  radius  of  a  circle  be  successively  inscribed 
as  a  chord  within  its  circumference,  after  the  sixth  inscription  we 
arrive  at  the  point  from  which  we  set  out,  and  they  concluded  from 
this  that  the  circumference  of  a  circle  must  be  a  little  larger  than  a 
line  which  is  six  times  as  long  as  the  radius,  that  is  three  times  as 
long  as  the  diameter.  A  trace  of  this  Babylonian  method  of  com- 
putation may  even  be  found  in  the  Bible;  for  in  i  Kings  vii.  23, 
and  2  Chron.  iv.  2,  the  great  laver  is  described,  which  under  the 
name  of  the  "molten  sea"  constituted  an  ornament  of  the  temple 
of  Solomon  ;  and  it  is  said  of  this  vessel  that  it  measured  ten  cubits 
from  brim  to  brim,  and  thirty  cubits  round  about.  The  number  3 
as  the  ratio  of  the  circumference  to  the  diameter  is  still  more  plainly 
given  in  the  Talmud,  where  we  read  that  "that  which  measures 
three  lengths  in  circumference  is  one  length  across." 

With  regard  to  the  earlier  Greek  mathematicians — as  Thales 
and  Pythagoras — we  know  that  they  acquired  their  elementary 
mathematical  knowledge  in  Egypt.  But  nothing  has  been  handed 
down  to  us  which  shows  that  they  knew  of  the  old  Egyptian  quad 
rature,  or  that  they  dealt  with  the  problem  at  all.  But  tradition 
says,  that,  subsequently,  the  teacher  of  Euripides  and  Pericles,  the 
great  philosopher  and  mathematician  Anaxagoras,  whom  Plato  so 
highly  praised,  "drew  the  quadrature  of  the  circle"  in  prison,  in  the 
year  434  B.  C.  This  is  the  account  of  Plutarch  in  the-seventeenth 
chapter  of  his  work  De  Exilic.  The  method  is  not  told  us  in  which 
Anaxagoras  is  supposed  to  have  solved  the  problem,  and  it  is  not 
said  whether,  knowingly  or  unknowingly,  he  gave  an  approximate 
solution  after  the  manner  of  Ahmes.  But  at  any  rate,  to  Anaxago- 
ras belongs  the  merit  of  having  called  attention  to  a  problem  that 
was  to  bear  rich  fruit  b}'  inciting  Grecian  scholars  to  busy  them- 
selves with  geometry,  and  thus  more  and  more  to  advance  that 
science. 

Again,  it  is  reported  that  the  mathematician  Hippias  of  Elis 
invented  a  curved  line  that  could  be  made  to  serve  a  double  pur- 
pose :  first,  to  trisect  an  angle,  and  second  to  square  the  circle. 
This   curved   line   is   the   TETpaywvt'^ouo-a  so  often  mentioned  b}'  the 


124  l'^^  SQUARING  OF  THE  CIRCLE. 

later  Greek  mathematicians,  and  by  the  Romans  called  "quadrat- 
rix. "  Regarding  the  nature  of  this  curve  we  have  exact  knowledge 
from  Pappus.  But  it  will  be  sufficient  here  to  state  that  the  quad- 
ratrix  is  not  a  circle  nor  a  portion  of  a  circle,  so  that  its  construc- 
tion is  not  possible  by  means  of  the  postulates  enumerated  in  the 
preceding  section.  And  therefore  the  solution  of  the  quadrature 
of  the  circle  founded  on  the  construction  of  the  quadratrix  is  not 
an  elementary  solution  in  the  sense  discussed  in  the  last  section. 
We  can,  it  is  true,  conceive  a  mechanism  that  will  draw  this  curve 
as  well  as  compasses  draw  a  circle ;  and  with  the  assistance  of  a 
mechanism  of  this  description  the  squaring  of  the  circle  is  solvable 
with  exactitude.  But  if  it  be  allowed  to  employ  in  a  solution  an 
apparatus  especially  adapted  thereto,  every  problem  may  be  said  to 
be  solvable.  Strictly  taken,  the  invention  of  the  curve  of  Hippias 
substitutes  for  one  insuperable  difficulty  another  equally  insuper- 
able. Some  time  afterwards,  about  the  year  350  B.  C,  the  mathe- 
matician Dinostratus  showed  that  the  quadratrix  could  also  be  used 
to  solve  the  problem  of  rectification,  and  from  that  time  on  this 
problem  plays  almost  the  same  role  in  Grecian  mathematics  as  the 
related  problem  of  quadrature. 

As  these  problems  gradually  became  known  to  the  non-math- 
ematicians of  Greece,  attempts  at  solution  at  once  sprang  up 
that  are  worthy  of  a  place  by  the  side  of  the  solutions  of  modern 
amateur  circle-squarers.  The  Sophists  especially  believed  them- 
selves competent  by  seductive  dialectic  to  take  the  stronghold  that 
had  defied  the  intellectual  onslaughts  of  the  greatest  mathemati- 
cians. With  verbal  nicety,  amounting  to  puerility,  it  was  said  that 
the  squaring  of  the  circle  depended  upon  the  finding  of  a  number 
which  represented  in  itself  both  a  square  and  a  circle  ;  a  square  by 
being  a  square  number,  a  circle  in  that  it  ended  with  the  same 
number  as  the  root  number  from  which,  by  multiplication  with  it- 
self, it  was  produced.  The  number  36,  accordingly,  was,  as  they 
thought,  the  one  that  embodied  the  solution  of  the  famous  prob- 
lem. 

Contrasted  with  this  twisting  of  words  the  speculations  of  Bry- 
son  and  Antiphon,  both   contemporaries  of  Socrates,  though  inex- 


THE  SQUARING  OF  THE  CIRCLE.  1 25 

act,  appear  in  a  high  degree  intelligent.  Antiphon  divided  the 
circle  into  four  equal  arcs,  and  by  joining  the  points  of  division  ob- 
tained a  square  ;  he  then  divided  each  arc  again  into  two  equal 
parts  and  thus  obtained  an  inscribed  octagon  ;  thence  he  constructed 
an  inscribed  i6-gon,  and  perceived  that  the  figure  so  inscribed 
more  and  more  approached  the  shape  of  a  circle.  In  this  way,  he 
said,  one  should  proceed,  until  there  was  inscribed  in  the  circle  a 
polygon  whose  sides  by  reason  of  their  smallness  should  coincide 
with  the  circle.  Now  this  polygon  could,  by  methods  already 
taught  by  the  Pythagoreans,  be  converted  into  a  square  of  equal 
area ;  and  upon  the  basis  of  this  fact  Antiphon  regarded  the. squar- 
ing of  the  circle  as  solved. 

Nothing  can  be  said  against  this  method  except  that,  however 
far  the  bisection  of  the  arcs  is  carried,  the  result  still  remains  an 
approximate  one. 

The  attempt  of  Bryson  of  Heraclea  was  better  still ;  for  this 
scholar  did  not  rest  content  with  finding  a  square  that  was  very 
little  smaller  than  the  circle,  but  obtained  by  means  of  circum- 
scribed pol3'gons  another  square  that  was  very  little  larger  than  the 
circle.  Only  Bryson  committed  the  error  of  believing  that  the  area 
of  the  circle  was  the  arithmetical  mean  between  an  inscribed  and  a 
circumscribed  pol3'gon  of  an  equal  number  of  sides.  Notwith- 
standing this  error,  however,  to  Bryson  belongs  the  merit — first,  of 
having  introduced  into  mathematics  by  his  emphasis  of  the  neces- 
sity of  a  square  which  was  too  large  and  one  which  was  too  small, 
the  conception  of  upper  and  lower  "limits"  in  approximations; 
and  secondly,  by  his  comparison  of  the  regular  inscribed  and  cir- 
cumscribed polygons  with  a  circle,  of  having  indicated  to  Archime- 
des the  way  by  which  an  approximate  value  of  n  was  to  be  reached. 

Not  long  after  Antiphon  and  Bryson,  Hippocrates  of  Chios 
treated  the  problem,  which  had  now  become  more  and  more  fa- 
mous, from  a  new  point  of  view.  Hippocrates  was  not  satisfied 
with  approximate  equalities,  and  searched  for  curvilinearly  bounded 
plane  figures  which  should  be  mathematically  equal  to  a  recti- 
linearly  bounded  figure,  and  which  therefore  could  be  converted  by 
straight  edge  and  compasses  into  a  square  equal  in  area.      First, 


\ 


126 


THE   SQUARING  OF  THE  CIRCLE. 


Hippocrates  found  that  the  crescent-shaped  plane  figure  produced 
by  drawing  two  perpendicular  radii  in  a  circle  and  describing  upon 
the  line  joining  their  extremities  a  semicircle,  is  exactly  equal  in 
area  to  the  triangle  that  is  formed  bj'  this  line  of  junction  and  the 
two  radii  ;  and  upon  the  basis  of  this  fact  the  endeavors  of  this  un- 
tiring scholar  were  directed  towards  converting  a  circle  into  a  cres- 
cent. Naturally  he  was  unable  to  attain  this  object,  but  by  his  ef- 
forts he  discovered  many  new  geometrical  truths  ;  among  others 
being  the  generalised  form  of  the  theorem  mentioned,  which  bears 
to  the  present  day  the  name  of  lunulae  Hippocratis,  the  lunes  of 
Hippocrates.  Thus,  in  the  case  of  Hippocrates,  it  appears  in  the 
plainest  light,  how  precisely  the  insolvable  problems  of  science  are 
qualified  to  advance  science ;  in  that  they  incite  investigators  to 
devote  themselves  with  persistence  to  its  study  and  thus  to  fathom 
its  utmost  depths. 

Following  Hippocrates  in  the  historical  line  of  the  great  Gre- 
cian geometricians  comes  the  S5'stematist  Euclid,  whose  rigid  form- 
ulation of  geometrical  principles  has  remained  the  standard  presen- 
tation down  to  the  present  century.  The  Elements  of  Euclid, 
•however,  contain  nothing  relating  to  the  quadrature  of  the  circle 
or  to  circle-computation.  Comparisons  of  surfaces  which  relate  to 
the  circle  are  indeed  found  in  the  work,  but  nowhere  a  computa- 
tion of  the  circumference  of  a  circle  or  of  the  area  of  a  circle.  This 
palpable  gap  in  Euclid's  system  was  filled  by  Archimedes,  the 
greatest  mathematician  of  antiquity. 

Archimedes  was  born  in  Syracuse  in  the  5'ear  287  B.  C,  and 
devoted  his  life,  which  was  spent  in  that  city,  to  the  mathematical 
and  the  physical  sciences,  which  he  enriched  with  invaluable  con- 
tributions. He  lived  in  Syracuse  till  the  taking  of  the  town  l)y 
Marcellus,  in  the  year  212  B.  C,  when  he  fell  by  the  hand  of  a  Ro- 
man soldier  whom  he  had  forbidden  to  destroy  the  figures  he  had 
drawn  in  the  sand.  To  the  greatest  performances  of  Archirnedes 
the  successful  computation  of  the  number  n  unquestionably  be- 
longs. Like  Bryson  he  started  with  regular  inscribed  and  circum- 
scribed polygons.  He  showed  how  it  was  possible,  beginning  with 
the  perimeter  of  an  inscribed  hexagon,  which  is  equal  to  six  radii, 


THE  SQUARING  OF  THE  CIRCLE.  I  27 

to  obtain  by  calculation  the  perimeter  of  a  regular  dodecagon,  and 
then  the  perimeter  of  a  figure  having  double  the  number  of  sides  of 
that,  and  so  on.  Treating,  then,  the  circumscribed  polygons  in  a 
similar  manner,  and  proceeding  with  both  series  of  polygons  up  to 
a  regular  g6-sided  polygon^  he  discovered  on  the  one  hand  that  the 
ratio  of  the  perimeter  of  the  inscribed  g6- sided  polygon  to  the 
diameter  was  greater  than  6336  :  2oiy}^,  and  on  the  other  hand,  that 
the  corresponding  ratio  with  respect  to  the  circumscribed  g6-sided 
polygon  was  smaller  than  14688:4673^.  He  inferred  from  this, 
that  the  number  tt,  the  ratio  of  the  circumference  to  the  diameter, 
was  greater  than  the  fraction  .y„:,^  and  smaller  than  -— — -.  Reducing 
the  two  limits  thus  found  for  the  value  of  tt,  Archimedes  then 
showed  that  the  first  fraction  was  greater  than  31^,  and  that  the 
second  fraction  was  smaller  than  3I,  whence  it  followed  with  cer- 
tainty that  the  value  sought  for  tt  lay  between  3^^  and  31^.  The 
larger  of  these  two  approximate  values  is  the  only  one  usually 
learned  and  employed.  That  which  fills  us  with  most  astonish- 
ment in  the  case  of  Archimedes's  computation  of  tt,  is,  first,  the 
great  acumen  and  accuracy  displayed  by  him  in  all  the  details  of 
the  computation,  and  secondly  the  unwearied  perseverance  which 
he  exercised  in  calculating  the  limits  of  tt  without  the  help  of  the 
Arabian  system  of  numerals  and  the  decimal  notation.  For  it  must 
be  considered  that  at  many  stages  of  the  computation  what  we  call 
the  extraction  of  roots  was  necessary,  and  that  Archimedes  could 
only  by  extremely  tedious  calculations  obtain  ratios  that  expressed 
approximately  the  roots  of  given  numbers  and  fractions.* 

With  regard  to  the  mathematicians  of  Greece  that  follow  Archi- 
medes, all  refer  to  and  employ  the  approximate  value  of  3I  for  tt, 
without,  how^ever,  contributing  anything  essentially  new  to  the 
problems  of  quadrature  and  of  cyclometry.  Thus  Hero  of  Alex- 
andria, the  father  of  surveying,  who  flourished  about  the  year 
100  B.  C,  employs  for  purposes  of  practical  measurement  some 


*For  Archimedes's  actual  researches,  see  Rudio,  Archimedes,  Htiygetis,  Lam- 
bert, Legetidrc,  vier  Abhand.  iiber  die  Krcismessung  (Leipsic,  1892),  where 
translations  of  the  works  of   these  four  authors  on  cyclometry  will  be  found.  —  Tr. 


128  THE   SQUARING  OF  THE  CIRCLE. 

times  the  value  34  for  n  and  sometimes  even  the  rougher  ap- 
proximation ;r  =  3.  The  astronomer  Ptolemy,  who  lived  in  Alex- 
andria about  the  year  150  A.  D.,  and  who  was  famous  as  being  the 
author  of  the  planetary  system  universally  recognised  as  correct 
down  to  the  time  of  Copernicus,  was  the  only  one  who  furnished  a 
more  exact  value ;  this  he  designated,  in  the  sexagesimal  sj'stem  of 
fractional  notation  which  he  employed,  by  3,  8,  30, — that  is  3  and 
-^^  and  ^f §0,  or  as  we  now  say  3  degrees,  8  minutes  {^partes  minutae 
primae),  and  30  seconds  {partes  nmiutae  secundae).  As  a  matter  of 
fact,  the  expression  3  +  5%  +  jf ft  ^  StVo  represents  the  number  n 
more  exactly  than  3^  ;  but  on  the  other  hand,  is,  by  reason  of  the 
magnitude  of  the  numbers  17  and  120  as  compared  with  the  num- 
bers I  and  7,  more  cumbersome. 

IV. 

THE    ROMANS,    HINDUS,    CHINESE,    ARABS,    AND   THE  CHRISTIAN 
NATIONS  TO  THE  TIME   OF  NEWTON. 

In  the  mathematical  sciences,  more  than  in  any  other,  the  Ro- 
mans stood  upon  the  shoulders  of  the  Greeks.  Indeed,  with  re- 
spect to  cyclometry,  they  not  only  did  not  add  anything  new  to  the 
Grecian  discoveries,  but  frequently  even  evinced  that  they  either 
did  not  know  of  the  beautiful  result  obtained  by  Archimedes,  or  at 
least  could  not  appreciate  it.  For  instance,  Vitruvius,  who  lived 
during  the  time  of  Augustus,  computed  that  a  wheel  4  feet  in  diam- 
eter must  measure  12^  feet  in  circumference;  in  other  words,  he 
made  n  equal  to  3^.  And,  similarly,  a  treatise  on  surveying,  pre- 
served to  us  in  the  Gudian  manuscript  of  the  library  of  Wolfen- 
biittel,  contains  the  following  instructions  for  squaring  the  circle  : 
Divide  the  circumference  of  a  circle  into  four  parts  and  make  one 
part  the  side  of  a  square  \  this  square  will  be  equal  in  area  to  the 
circle.  Apart  from  the  fact  that  the  rectification  of  the  arc  of  a 
circle  is  requisite  to  the  construction  of  a  square  of  this  kind,  the 
Roman  quadrature,  viewed  as  a  calculation,  is  more  inexact  even 
than  any  other  known  computation  ;  for  its  result  is  that  ;r  =  4. 

The  mathematical  performances  of  the  Hindus  were  not  only 
greater  than  those  of  the   Romans,  but  in  certain  directions  sur- 


THE   SQUARING  OF  THE  CIRCLE.  I2g 

passed  even  those  of  the  Greeks.  In  the  most  ancient  source  of 
the  mathematics  of  India  that  we  know  of,  the  Culvasutras,  which 
date  back  to  a  little  before  our  chronological  era,  we  do  not  find,  it 
is  true,  the  squaring  of  the  circle  treated  of,  but  the  opposite  prob- 
lem is  dealt  with,  which  might  fittingly  be  termed  the  circling  of 
the  square.  The  half  of  the  side  of  a  given  square  is  prolonged  in 
length  one  third  of  the  excess  of  half  the  diagonal  over  half  the 
side,  and  the  line  thus  obtained  is  taken  as  the  radius  of  the  circle 
equal  in  area  to  the  square.  The  simplest  way  to  obtain  an  idea 
of  the  exactness  of  this  construction  is  to  compute  how  great  n 
would  have  to  be  if  the  construction  were  exactly  correct.  We 
find  out  in  this  way  that  the  value  of  n  upon  which  the  Indian  cir- 
cling of  the  square  is  based,  is  about  from  five  to  six  hundredths 
smaller  than  the  true  value,  whereas  the  approximate  n  of  Archi- 
medes, 3I,  is  only  from  one  to  two  thousandths  too  large,  and  that 
the  old  Egyptian  value  exceeds  the  true  value  by  from  one  to  two 
hundredths. 

Cyclometry  very  probably  made  great  advances  among  the 
Hindus  in  the  first  four  or  five  centuries  of  our  era  ;  for  Aryabhatta, 
who  lived  about  the  year  500  after  Christ,  states,  that  the  ratio  of 
the  circumference  to  the  diameter  is  62832  :  20000,  an  approxima- 
tion that  in  exactness  surpasses  even  that  of  Ptolemy.  The  Hindu 
result  gives  3-1416  for  ;r,  while  n  really  lies  between  3-141592  and 
3 -14 1593.  How  the  Hindus  obtained  this  excellent  value  is  told  by 
Ganeca,  the  commentator  of  Bhaskara,  an  author  of  the  twelfth 
century.  Ganeca  says  that  the  method  of  Archimedes  was  carried 
still  farther  by  the  Hindu  mathematicians  ;  that  by  continually 
doubling  the  number  of  sides  they  proceeded  from  the  hexagon  to 
a  polygon  of  384  sides,  and  that  by  the  comparison  of  the  circum- 
ferences of  the  inscribed  and  circumscribed  384-sided  polygons  they 
found  that  n  was  equal  to  3927:1250.  It  will  be  seen  that  the  value 
given  By  Bhaskara  is  identical  with  the  value  of  Aryabhatta.  It  is 
further  worthy  of  remark  that  the  earlier  of  these  two  Hindu  math- 
ematTGians  does  not  mention  either  the  value  3I  of  Archimedes  or 
the  -(lalue  Sy^g-  of  Ptolemy,  but  that  the  later  one  knows  of  both 
values  and  especially  recommends  that  of  Archimedes  as  the  most 


130  THE   SQUARING  OF  THE  CIRCLE. 

useful  for  practical  applications.  Strange,  to  say,  the  good  ap- 
proximate value  of  Aryabhatta  does  not  occur  in  Brahmagupta,  the 
great  Hindu  mathematician  who  flourished  in  the  beginning  of  the 
seventh  century  ;  but  we  find  the  curious  information  in  this  author 
that  the  area  of  a  circle  is  exactly  equal  to  the  square  root  of  10 
when  the  radius  is  unity.  The  value  of  n  as  derivable  from  this 
formula, — a  value  from  two  to  three  hundredths  too  large, — has 
unquestionably  arisen  on  Hindu  soil.  For  it  occurs  in  no  Grecian 
mathematician  ;  and  Arabian  authors,  who  were  in  a  better  position 
than  we  to  know  Greek  and  Hindu  mathematical  literature,  declare 
that  the  approximation  which  makes  n  equal  to  the  square  root  of 
10,  is  of  Hindu  origin.  It  is  possible  that  the  Hindu  people,  who 
were  addicted  more  than  any  other  to  numeral  mysticism,  sought 
to  find  in  this  approximation  some  connection  with  the  fact  that 
man  has  ten  fingers,  and  that  accordingly  ten  is  the  basis  of  their 
numeral  system. 

Reviewing  the  achievements  of  the  Hindus  generally  with  re- 
spect to  the  problem  of  quadrature,  we  are  brought  to  recognise 
that  this  people,  whose  talents  lay  more  in  the  line  of  arithmetical 
computation  than  in  the  perception  of  spatial  relations,  accom- 
plished as  good  as  nothing  on  the  purely  geometrical  side  of  the 
problem,  but  that  the  merit  belongs  to  them  of  having  carried  the 
Archimedean  method  of  computing  n  several  stages  farther,  and  of 
having  obtained  in  this  way  a  much  more  exact  value  for  it — a  cir- 
cumstance that  is  explainable  when  we  consider  that  the  Hindus 
are  the  inventors  of  our  present  system  of  numeral  notation,  pos- 
sessing which  they  easily  outdid  Archimedes,  who  employed  the 
awkward  Greek  system. 

With  regard  to  the  Chinese,  this  people  operated  in  ancient 
times  with  the  Bab3lonian  value  for  n,  ox  3  ;  but  they  j^ossciStd 
knowledge  of  the  approximate  value  of  Archimedes  at  least  since 
the  end  of  the  sixth  century.  Besides  this,  there  appears  in*a  num- 
ber of  Chinese  mathematical  treatises  an  approximate  value  pe- 
culiarly their  own,  in  whicli  7r  =  3g\;  a  value,  however,  which  not- 
witlistanding  it  is  written  in  larger  figures,  is  no  better  than  that  of 


THE   SQUARING  OF  THE   CIRCLE.  I3I 

Archimedes.  Attempts  at  the  constructive  quadrature  of  the  circle 
are  not  found  among  the  Chinese. 

Greater  were  the  merits  of  the  Arabians  in  the  advancement  of 
mathematics ;  and  especially  in  virtue  of  the  fact  that  they  pre- 
served from  oblivion  the  results  of  bath  Greek  and  Hindu  research 
and  handed  them  down  to  the  Christian  countries  of  the  West.  The 
Arabians  expressly  distinguished  between  the  Archimedian  approx- 
imate value  and  the  two  Hindu  values,  the  square  root  of  lo  and 
the,  ratio  62832:20000.  This  distinction  occurs  also  in  Muhammed 
Ibri  Musa  Alchwarizmi,  ,the  same  scholar  who  in  the  beginning  of 
the.  ninth  century  brought  t-he  principles  of  our  present  system  of 
numerical  notation  from  ladia  and  introduced  it  into  the  Moham- 
medan world.  The  Arabians,  however,  did  not  study  the  numerical 
quadrature  of  the  ciircle  only,  but  also  the  constructive  ;  for  in- 
stance, an  attempt  of  this  kind  was  made  by  Ibn  Alhaitam,  who 
lived  in  Egypt  .about  the  year  1000  and  whose  treatise  upon  the 
squaring. 'of.  the  circle  is  preserved  in  a  Vatican  codex,  which  un- 
fortunately has  not  yet  been  edited. 

Christian  civilisation,  to  which  we  are  now  about  to  pass,  pro- 
druced  up  to  the  second  half  of  the  fifteenth  century  extremely  in- 
' significant  results  in  mathematics.  Even  with  regard  to  our  pres- 
ent problem  we  have  but  a  single  important  work  to  mention;  the 
work,  namely,  of  Frankos  von  Liittich  on  the  squaring  of  the  circle, 
.published  in  six  books,  but  preserved  only  in  fragments.  The 
author,  who  lived  in  the  first  half  of  the  eleventh  century,  was 
probably  a  pupil  of  Pope  Sylvester  H.,  who  was  himself  a  not  in- 
considerable mathematician  for  his  time  and  the  author  of  the  most 
celebrated  geometrical  treatise  of  the  period. 

Greater  interest  came  to  be  bestowed  upon  mathematics,  and 
especially  on  the  problem  of  the  quadrature  of  the  circle,  in  the 
second  half  of  the  fifteenth  century,  when  the  sciences  again  began 
to  revive.  This  interest  was  principally  aroused  by  Cardinal  Nico- 
las de  Cusa,  a  man  highly  esteemed  for  his  astronomical  and  calen- 
darial  studies.  He  claimed  to  have  discovered  the  quadrature  of 
the  circle  by  employing  only  straight  edge  and  compasses  and  thus 
attracted  the  attention  of  scholars  to  the  historic  problem.     People 


132  THE   SQUARING  OF  THE  CIRCLE. 

believed  the  famous  Cardinal,  and  marvelled  at  his  wisdom,  until 
Regiomontanus,  in  letters  written  in  1464  and  1465  and  published 
in  1533,  rigorously  demonstrated  that  the  Cardinal's  quadrature 
was  incorrect.  The  construction  of  Cusa  was  as  follows.  The  ra- 
dius of  a  circle  is  prolonged  a  distance  equal  to  the  side  of  the  in- 
scribed square;  the  line  so. obtained  is  taken  as  the  diameter  of  a 
second  circle,  and  in  the  latter  an  equilateral  triangle  is  described ; 
then  the  perimeter  of  the  latter  is  equal  to  the  circumference  of  the 
original  circle.  If  this  construction,  which  its  inventor  regarded  as 
exact,  be  considered  as  a  construction  of  .approximation,  it  will  be 
found  to  be  more  inexact  even  than  the  construction  resulting  from 
the  value  ;r  =  3i.  For  by  Cusa's  method  tt  would  be  from  five  to 
six  thousandths  smaller  than  it  really  is.  ,  > 

In  the  beginning  of  the  sixteenth  century  ascertain  BovillillB 
appears,  who  also  gave  the  construction  of  Cusa,— this  time  with- 
out notice.  But  about  the  middle  of  the  sixteenth  century  a  book 
was  published  which  the  scholars  of  the  time  at  first  received.- with 
interest.  It  bore  the  proud  title  De  Rebus  MaihemafiH^  HacieriMts- 
Desideratis.  Its  author,  Orontius  Finaeus,  represented  that  he  ha^' ' 
overcome  all  the  difficulties  that  had  ever  stood  in  the  way  of  geo- 
metrical investigators  ;  and  incidentally  he  also  communicated  to 
the  world  the  "true  quadrature"  of  the  circle.  His  fame  was  siiort- 
lived.  For  soon  afterwards,  in  a  book  entitled  De  Erratis  Oroiiiii,  .. 
the  Portuguese  Petrus  Nonius  demonstrated  that  Orontius's  quad- 
rature, like  most  of  his  other  professed  discoveries,  was  incorrect. 

In  the  succeeding  period  the  number  of  circle-squarers  so  in.'-  v 
creased  that  we  shall  have  to  limit  ourselves  to  those  whom  mathe- 
maticians recognise.  And  particularly  is  Simon  Van  Eyck  to  be 
mentioned,  who  towards  the  close  of  the  sixteenth  century  pub- 
lished a  quadrature  which  was  so  approximate  that  the  value  of  Tt 
derived  from  it  was  more  exact  even  than  that  of  Archimedes;  and  . 
to  disprove  it  the  mathematician  Peter  Metius  was  obliged  to  seek 
a  still  more  accurate  value  than  3I.  The  erroneous  quadrature  of 
Van  Eyck  was  thus  the  occasion  of  Metius's  discovery  that  the  ra- 
tio 355:113,  or  31^"^,  varied  from  the  true  value  of  n  by  less  than 
one   one-millionth,   eclipsing    accordingly^   all   values    hitherto    ob- 


THE  SQUARING  OF  THE  CIRCLE.  I  33 

tained.  Moreover,  it  is  demonstrable  by  the  theory  of  continued 
fractions,  that,  admitting  figures  to  four  places  only,  no  two  num- 
bers more  exactly  represent  the  value  of  n  than  355  and  113. 

In  the  same  way  the  quadrature  of  the  great  philologist  Joseph 
Scaliger  led  to  refutations.  Like  most  circle-squarers  who  believe 
in  their  discovery,  Scaliger  also  was  little  versed  in  the  elements  of 
geometry.  He  solved  the  famous  problem,  however, — at  least  in 
his  own  opinion, — and  published  in  1592  a  book  upon  it,  which 
bore  the  pretentious  title  Nova  Cyclometria,  and  in  which  the  name 
of  Archimedes  was  derided.  The  baselessness  of  his  supposed  dis- 
covery was  demonstrated  to  him  by  the  greatest  mathematicians  of 
his  time ;  namely,  Vieta,  Adrianus  Romanus,  and  Clavius. 

Of  the  erring  circle-squarers  that  flourished  before  the  middle 
of  the  seventeenth  century  three  others  deserve  particular  mention, 
— Longomontanus  of  Copenhagen,  who  rendered  such  great  ser- 
vices to  astronomy,  the  Neapolitan  John  Porta,  and  Gregory  of  St. 
Vincent.  Longomontanus  made  tt  =  3yyjy8_5_^  and  was  so  convinced 
of  the  correctness  of  his  result  as  to  thank  God  fervently,  in  the 
preface  to  his  work  Invcntio  Quadraturae  Circuli,  that  He  had 
granted  him  in  his  old  age  the  strength  to  conquer  the  celebrated 
difficulty.  John  Porta  followed  the  example  of  Hippocrates  and 
endeavored  to  solve  the  problem  by  a  comparison  of  lunes.  Gregory 
of  St.  Vincent  published  a  quadrature,  the  error  of  which  was  very 
hard  to  detect  but  was  finally  discovered  by  Descartes. 

Of  the  famous  mathematicians  who  dealt  with  our  problem  in 
the  period  between  the  close  of  the  fifteenth  century  and  the  time 
of  Newton,  we  first  meet  with  Peter  Metius,  before  mentioned,  who 
succeeded  in  finding  in  the  fraction  355:113  the  best  approximate 
value  for  n  involving  small  numbers  only.  The  problem  received 
a  different  advancement  at  the  hands  of  the  famous  mathematician 
Vieta.  Vieta  was  the  first  to  whom  the  idea  occurred  of  represent- 
ing 71  with  mathematical  exactness  by  an  infinite  series  of  definitely 
prescribed  operations.  By  comparing  inscribed  and  circumscribed 
polygons,  Vieta  found  that  we  approach  nearer  and  nearer  to  n  if 
we  cause  the  operations  of  extracting  the  square  root  of  \,  and 
certain  related  additions  and  multiplications,  to  succeed  each  other 


134  "^HE   SQUARING  OF  THE   CIRCLK. 

in  a  certain  manner,  and  that  n  must  come  out  exactly,  if  this  series 
of  operations  could  be  continued  indefinitely.  ,  Vieta  thus  found 
that  to  a  diameter  of  loooo  million  units  a  circumference  belongs  of 
from  31415  million  926535  units  to  31415  million  926536  units  of 
the  same  length. 

But  Vieta  was  outdone  b}'  the  Netherlander  Adrianus  Romanus, 
who  added  five  additional  decimal  places  to  the  ten  of  Vieta.  To 
accomplish  this  he  computed  with  unspeakable  labor  the  circum- 
ference of  a  regular  circumscribed  polygon  of  1073741824  sides. 
This  number  is  the  thirtieth  power  of  2.  Yet  great  as  the  labor  of 
Adrianus  Romanus  was,  that  of  Ludolf  Van  Ceulen  was  still  greater;* 
for  the  latter  calculator  succeeded  in  carrying  the  Archimedean 
process  of  approximation  for  the  value  of  n  to  35  decim.al  places  ; 
that  is,  the  deviation  from  the  true  value  was  smaller  than  one  one- 
thousand  quintillionth,  a  degree  of  exactness  that  we  can  have 
scarcely  any  conception  of.  Ludolf  published  the  figures  of  the 
tremendous  computation  that  led  to  his  result.  His  calculation 
was  carefully  examined  by  the  mathematician  Griemberger  and  de- 
clared to  be  correct.  Ludolf  was  justly  proud  of  his  work,  and  fol- 
lowing the  example  of  Archimedes,  requested  in  his  will  that  the 
result  of  his  most  important  mathematical  performance,  the  com- 
putation of  71  to  35  decimal  places,  be  engraved  upon  his  tomb- 
stone ;  a  request  which  is  said  to  have  been  carried  out.  \\\  honor 
of  Ludolf,  n  is  called  to-day  in  Germany  the  Ludolfian  number. 

Although  through  the  labor  of  Ludolf  a  degree  of  exactness  for 
cyclometrical  operations  was  now  obtained  that  was  more  than  suf- 
ficient for  any  practical  purpose  that  could  ever  arise,  neither  the 
problem  of  constructive  rectification  nor  that  of  constructive  quad- 
rature had  been  in  any  respect  theoretically  advanced  thereby.  The 
investigations  conducted  by  the  famous  mathematicians  and  phys- 
icists Huygens  and  Snell  about  the  middle  of  the  seveateenth  cen- 
tury, were  more  important  from  a  mathematical  point  of  view  than 
the  work  of  Ludolf.  In  his  book  Cyclornetricus  Snell  took  the  posi- 
tion that  the  method  of  comparison  of  polygons,  which  originated 
with  Archimedes  and  was  employed  by  Ludolf,  was  not  necessarily 
tlie  best  method  of  attaining  the  end  sought  ;   and  he  succeeded   by 


THE  SQUARING  Ol    THE  CIRCLF.  135 

employing  propositions  which  state  that  certain  arcs  of  a  circle  are 
greater  or  smaller  than  certain  straight  lines  connected  with  the 
circle,  in  obtaining  methods  that  make  it  possible  to  reach  results 
like  the  Ludolfian  with  much  less  labor  of  calculation.  The  beau- 
tiful theorems  of  Snell  were  proved  a  second  time,  and  better 
proved,  by  the  celebrated  Dutch  promoter  of  the  science  of  optics, 
Huygens  {Opera  Varia,  p.  365  et  seq. ;  T/icore//iata  De  Circuli  et 
Hyperbolae  Qiiadratura,  1651),  as  well  as  perfected  in  many  ways 
by  him.  Snell  and  Huygens  were  full}'  aware  that  they  had  ad- 
vanced the  problem  of  numerical  quadrature  only,  and  not  that  of 
the  constructive  quadrature.  This  plainly  appeared  in  Huygens's 
case  from  the  vehement  dispute  which  he  conducted  with  the  Eng- 
lish mathematician  James  Gregory.  This  controversy  is  significant 
for  the  history  of  our  problem,  from  the  fact  that  Gregory  made 
the  first  attempt  to  prove  that  the  squaring  of  the  circle  with  straight 
edge  and  compasses  was  impossible.  The  result  of  the  contro- 
versy, to  which  we  owe  many  valuable  tracts,  was,  that  Huygens 
finally  demonstrated  in  an  incontrovertible  manner  the  incorrect- 
ness of  Gregory's  proof  of  impossibility,  adding  that  he  also  was  of 
opinion  that  the  solution  of  the  problem  with  straight  edge  and 
compasses  was  impossible,  but  nevertheless  was  not  himself  able 
to  demonstrate  this  fact.  And  Newton  later  expressed  himself  to 
the  same  effect.  As  a  matter  of  fact  a  period  of  over  200  years 
elapsed  before  higher  mathematics  was  far  enough  advanced  to 
furnish  a  rigorous  demonstration  of  impossibility. 

V. 

FROM  NEWTON  TO  THE  PRESENT. 

Before  we  proceed  to  consider  the  promotive  influence  which 
the  invention  of  the  differential  and  the  integral  calculus  exercised 
upon  our  problem,  we  shall  enumerate  a  few  at  least  of  that  never- 
ending  succession  of  erring  quadrators  who  delighted  the  world 
with  the  products  of  their  ingenuity  from  the  time  of  Newton  to 
the  present  ;  and  out  of  a  pious  and  sincere  regard  for  the  contem- 
porary world,  we  shall  omit  entirely  to  speak  of  the  circle-squarers 
«f  our  own  time. 


13b  THE  SQUARING  OF  THE  CIRCLE. 

First  to  be  mentioned  is  the  celebrated  English  philosopher 
Hobbes.  In  his  book  De  Problematis  Fhysicis,  in  which  he  pro- 
poses to  explain  the  phenomena  of  gravity  and  of  ocean  tides,  he 
also  takes  up  the  quadrature  of  the  circle  and  gives  a  very  trivial 
construction,  which  in  his  opinion  definitively  solved  the  problem. 
It  made  7r  =  3|.  In  view  of  Hobbes's  importance  as  a  philosopher, 
two  mathematicians,  Huygens  and  Wallis,  thought  it  proper  to 
refute  him  at  length.  But  Hobbes  defended  his  position  in  a  spe- 
cial treatise,  where  to  sustain  at  least  the  appearance  of  being  right, 
he  disputed  the  fundamental  principles  of  geometry  and  the  the- 
orem of  Pythagoras. 

In  the  last  century  France  especially  was  rich  in  circle-squarers. 
We  will  mention  :  Oliver  de  Serres,  who  by  means  of  a  pair  of 
scales  determined  that  a  circle  weighed  as  much  as  the  square  upon 
the  side  of  the  equilateral  triangle  inscribed  in  it,  that  therefore 
they  must  have  the  same  area,  an  experiment  in  which  ;t=3; 
Mathulon,  who  offered  in  legal  form  a  reward  of  a  thousand  dol- 
lars to  the  person  who  would  point  out  an  error  in  his  solution  of 
the  problem,  and  who  was  actually  compelled  by  the  courts  to  pay 
the  money  ;  Basselin,  who  believed  that  his  quadrature  must  be 
right  because  it  agreed  with  the  approximate  value  of  Archimedes, 
and  who  anathematised  his  ungrateful  contemporaries,  in  the  con- 
fidence that  he  would  be  recognised  by  posterity ;  Liger,  who 
proved  that  a  part  is  greater  than  the  whole  and  to  whom  therefore 
the  quadrature  of  the  circle  was  child's  play  ;  Clerget,  who  based 
his  solution  upon  the  principle  that  a  circle  is  a  polygon  of  a  defi- 
nite number  of  sides,  and  who  calculated,  also,  among  other  things, 
how  large  the  point  is  at  which  two  circles  touch. 

Germany  and  Poland  also  furnish  their  contingent  to  the  army 
of  circle-squarers.  Lieutenant-Colonel  Corsonich  produced  a  quad- 
rature in  which  7t  equalled  3^,  and  promised  fifty  ducats  to  the  per- 
son who  could  prove  that  it  was  incorrect.  Hesse  of  Berlin  wrote 
an  arithmetic  in  1776,  in  which  a  true  quadrature  was  also  "made 
known,"  n  being  exactly  equal  to  3^A.  About  the  same  time  Pro- 
fessor Bischoff  of  Stettin  defended  a  quadrature  previously  pub- 
lished by  Captain  Leistner,   Preacher  Merkel,   and  Schoolmaster 


THE  SQUARING  OF  THE  CIRCLE.  1 37 

Bohm,  which  virtually  made  n  equal  to  the  square  of  ||,  not  even 
attaining  the  approximation  of  Archimedes. 

From  attempts  of  this  character  are  to  be  clearly  distinguished 
constructions  of  approximation  in  which  the  inventor  is  aware  that 
he  has  not  found  a  mathematically  exact  construction,  but  only  an 
approximate  one.  The  value  of  such  a  construction  will  depend 
upon  two  things — first,  upon  the  degree  of  exactness  with  which  it 
is  numerically  expressed,  and  secondly  on  whether  the  construc- 
tion can  be  easily  made  with  straight  edge  and  compasses.  Con- 
structions of  this  kind,  simple  in  form  and  yet  sufficiently  exact  for 
practical  purposes,  have  been  produced  for  centuries  in  great  num- 
bers. The  great  mathematician  Euler,  who  died  in  1783,  did  not 
think  it  out  of  place  to  attempt  an  approximate  construction  of  this 
kind.  A  very  simple  construction  for  the  rectification  of  the  circle 
and  one  which  has  passed  into  many  geometrical  text-books  is  that 
published  by  Kochansky  in  1685  in  the  Leipziger  Berichte.  It  is  as 
follows:  "Erect  upon  the  diameter  of  a  circle  at  its  extremities 
perpendiculars ;  with  the  centre  as  vertex  and  the  diameter  as  side 
construct  an  angle  of  30°;  find  the  point  of  intersection  of  the 
line  last  drawn  with  the  perpendicular,  and  join  this  point  of  inter- 
section with  that  point  on  the  other  perpendicular  which  is  dis- 
tant three  radii  from  the  base  of  the  perpendicular.  The  line  of 
junction  so  obtained  is  very  approximately  equal  to  one-half  of  the 
circumference  of  the  given  circle."  Calculation  shows  that  the  dif- 
ference between  the  true  length  of  the  circumference  and  the  line 
thus  constructed  is  less  than  j^^VnTr  °^  ^^®  diameter. 

Although  such  constructions  of  approximation  are  very  inter- 
esting in  themselves,  they  nevertheless  play  but  a  subordinate  role 
in  the  history  of  the  squaring  of  the  circle ;  for  on  the  one  hand 
they  can  never  furnish  greater  exactness  for  circle-computation 
than  the  thirty-five  decimal  places  which  Ludolf  found,  and  on  the 
other  hand  they  are  not  adapted  to  advance  in  any  way  the  ques- 
tion whether  the  exact  quadrature  of  the  circle  with  straight-edge 
and  compasses  is  possifcle. 

The  numerical  side  of  the  problem,  however,  was  considerably 
advanced  by  the  new  mathematical  methods  perfected  by  Newton 


138  THE  SQUARING  OF  THE  CIRCLE. 

and  Leibnitz,  and  known  as  the  differential  and  the  integral  cal- 
culus. 

About  the  middle  of  the  seventeenth  century,  before  Newton 
and  Leibnitz  represented  tt  by  series  of  powers,  the  English  mathe- 
maticians Wallis  and  Lord  Brouncker,  Newton's  predecessors  in 
certain  lines,  succeeded  in  representing  tt  by  an  infinite  series  of 
figures  combined  according  to  the  first  four  rules  of  arithmetic.  A 
new  method  of  computation  was  thus  opened.  Wallis  found  that 
the  fourth  part  of  tt  is  represented  by  the  regularly  formed  produc: 

more  and  more  exactly  the  farther  the  multiplication  is  continued, 
and  that  the  result  always  comes  out  too  small  if  we  stop  at  a  proper 
fraction  but  too  large  if  we  stop  at  an  improper  fraction.  Lord 
Brouncker,  on  the  other  hand,  represents  the  value  in  question  by 
a  continued  fraction  in  which  the  denominators  are  all  2  and  the 
numerators  are  the  squares  of  the  odd  numbers.  Wallis,  to  whom 
Brouncker  had  communicated  his  elegant  result  without  proof,  de- 
monstrated the  same  in  his  AritJunetic  of  Infinites. 

The  computation  of  n  could  scarcely  have  been  pushed  to  a 
greater  degree  of  exactness  by  these  results  than  that  to  which  Lu- 
dolf  and  others  had  carried  it  by  the  older  and  more  laborious 
methods.  But  the  series  of  powers  derived  from  the  differential 
calculus  of  Newton  and  Leibnitz  furnished  a  means  of  computing 
7t  to  hundreds  of  decimal  places. 

Gregory,  Newton,  and  Leibnitz  found  that  the  fourth  part  of 
7t  was  equal  exactly  to 

T 1-1-1 1-1-1 1 U   1  

if  we  conceive  this  series,  which  is  called  the  Leibnitz  series,  con- 
tinued indefinitely.  This  series  is  wonderfully  simple  but  is  not 
adapted  to  the  computation  of  n,  for  the  reason  that  entirely  too 
many  members  have  to  be  taken  into  account  to  obtain  tc  accurately 
to  a  few  decimal  places  only.  The  original  formula,  however,  from 
which  this  series  is  derived,  gives  other  formulae  which  are  excel- 
lently adapted  to  the  actual  computation.  •The  original  formula  is 
the  general  series : 


THE   SQUARING  OF  THE  CIRCLE.  I39 

where  a  is  the  length  of  the  arc  belonging  to  any  central  angle  in  a 
circle  of  radius  i,  and  a  the  tangent  to  this  angle.  From  this  we 
derive  the  following  : 

+  iC'^'  +  ^'  +  ^'+-  •  •;  — ••  •» 

where  a,  b,  c  .  .  .  are  the  tangents  of  angles  whose  sum  is  45°.  De- 
termining, therefore,  the  values  oi  a,  I?,  c .  .  .  ,  which  are  equal  to 
small  and  convenient  fractions  and  fulfil  the  conditions  just  men- 
tioned, we  obtain  series  of  powers  which  are  adapted  to  the  com- 
putation of  7t. 

The  first  to  add  by  the  aid  of  series  of  this  description  addi- 
tional decimal  places  to  the  old  35  in  the  number  n  was  the  Eng- 
lish arithmetician  Abraham  Sharp,  who,  following  Halley's  instruc- 
tions, in  1700  worked  out  n  to  72  decimal  places.  A  little  later 
Machin,  professor  of  astronomy  in  London,  computed  n  to  100 
decimal  places,  by  putting,  in  the  series  given  above,  a^=bz=c  =  d 
=  1  and  <?=  —  T^l-g-,  that  is,  by  employing  the  following  series: 


7t 


L5         3-5'        5-5'        7-5^ 


I  I  ,    I 


239       3-239         5-239 

In  the  year  1819,  Lagny  of  Paris  outdid  the  computation  of 
Machin,  determining  in  two  different  ways  the  first  127  decimal 
places  of  TT.  Vega  then  obtained  as  many  as  140  places,  and  the 
Hamburg  arithmetician  Zacharias  Dase  went  as  far  as  200  places. 
The  latter  did  not  use  Machin's  series  in  his  calculation,  but  the 
series  produced  by  putting  in  the  general  series  above  given  a  =  ^, 
<^  =  |,  <r=:^.  Finally,  at  a  recent  date,  tt  has  been  computed  to 
500  places.* 

The  computation  to  so  many  decimal  places  may  serve  as  an 
illustration  of  the  excellence  of  the  modern  methods  as  contrasted 
with  those  anciently  employed,  but  it  has  otherwise  neither  a  theo- 

*In  1873  the  approximation  was  carried  by  Shanks  to  707  places  of  decimals. 
—  Trans. 


140  THE  SQUARING  OF  THE  CIRCLE. 

retical  nor  a  practical  value.  That  the  computation  of  n  to  saj-  15 
decimal  places  more  than  sufficiently  satisfies  the  subtlest  require- 
ments of  practice  may  be  gathered  from  a  concrete  example  of  the 
degree  of  exactness  thus  obtainable.  Imagine  a  circle  to  be  de- 
scribed with  Berlin  as  centre,  and  the  circumference  to  pass  through 
Hamburg  ;  then  let  the  circumference  of  the  circle  be  computed  by 
multiplying  its  diameter  by  the  value  of  ;r  to  15  decimal  places, 
and  then  conceive  it  to  be  actually  measured.  The  deviation  from 
the  true  length  in  so  large  a  circle  as  this  even  could  not  be  as  great 
as  the  18  millionth  part  of  a  millimetre. 

An  idea  can  hardly  be  obtained  of  the  degree  of  exactness  pro- 
duced by  100  decimal  places.  But  the  following  example  may  pos- 
sibly give  us  some  conception  of  it.  Conceive  a  sphere  constructed 
with  the  earth  as  centre,  and  imagine  its  surface  to  pass  through 
Sirius,  which  is  134^  millions  of  millions  of  kilometres  distant  from 
the  earth.  Then  imagine  this  enormous  sphere  to  be  so  packed 
with  microbes  that  in  every  cubic  millimetre  millions  of  millions  of 
these  diminutive  animalcula  are  present.  Now  conceive  these  mi- 
crobes to  be  all  unpacked  and  so  distributed  singly  along  a  straight 
line,  that  every  two  microbes  are  as  far  distant  from  each  other  as 
Sirius  from  us,  that  is  134^^  million  million  kilometres.  Conceive 
the  long  line  thus  fixed  by  all  the  microbes,  as  the  diameter  of  a 
circle,  and  imagine  the  circumference  of  it  to  be  calculated  by  mul- 
tiplying its  diameter  by  n  to  100  decimal  places.  Then,  in  the 
case  of  a  circle  of  this  enormous  magnitude  even,  the  circumference 
so  calculated  would  not  vary  from  the  real  circumference  by  a  mil- 
lionth part  of  a  millimetre. 

This  example  will  suffice  to  show  that  the  calculation  of  n  to 
100  or  500  decimal  places  is  wholly  useless. 

Before  we  close  this  chapter  upon  the  evaluation  of  n,  we 
must  mention  the  method,  less  fruitful  than  curious,  which  Profes- 
sor Wolff  of  Zurich  employed  some  decades  ago  to  compute  the 
value  of  ;r  to  3  places.*  The  floor  of  a  room  is  divided  up  into  equal 
squares,  so  as  to  resemble  a  huge  chess-board,  and  a  needle  ex- 

*See  also  A.  De  Morgan,  A  Budget  of  Paradoxes,  pp.  169-171. —  7'r. 


THE  SQUARING  OF  THE  CIRCLE.  I4I 

actly  equal  in  length  to  the  side  of  each  of  these  squares,  is  cast 
haphazard  upon  the  floor.  If  we  calculate,  now,  the  probabilities 
of  the  needle  so  falling  as  to  lie  wholly  within  one  of  the  squares, 
that  is  so  that  it  does  not  cross  any  of  the  parallel  lines  forming  the 
squares,  the  result  of  the  calculation  for  this  probability  will  be 
found  to  be  exactly  equal  to  n — 3.  Consequently,  a  sufficient 
number  of  casts  of  the  needle  according  to  the  law  of  large  num- 
bers must  give  the  value  of  n  approximately.  As  a  matter  of  fact, 
Professor  Wolff,  after  loooo  trials,  obtained  the  value  of  n  correctly 
to  3  decimal  places.  ^^---'^ 

Fruitful  as  the  calculus  of  Newton  and  Leibnitz  was  for  the        \ 
evaluation   of  n,  the  problem   of  converting  a  circle  into  a  square         \ 
having  exactly  the  same  area  was  in  no  wise  advanced  therebyT" 
Wallis,  Newton,  Leibnitz,  and  their  immediate  followers  distinctly 
recognised  this.     The  quadrature  of  the  circle  could  not  be  solved  ; 
but  it  also  could  not  be  proved  that  the  problem  was  insolvable 
with  straight  edge  and  compasses,  although  everybody  was  con- 
vinced of  its  insolvability.      In  mathematics,  however,  a  convictiojti. 
is  only  justified  when  supported  by  incontrovertible  proof ;  and  in 
the  place  of  endeavors  to  solve  the  quadrature  there  accordingly 
now  come  endeavors  to  prove  the  impossibility  of  solving  the  cele- 
brated problem. 

The  first  step  in  this  direction,  small  as  it  was,  was  made  by 
the  French  mathematician  Lambert,  who  proved  in  the  year  1761 
that  Tt  was  neither  a  rational  number  nor  even  the  square  root  of  a 
rational  number  ;  that  is,  that  neither  n  nor  the  square  of  n  could 
be  exactly  represented  by  a  fraction  the  denominator  and  numera- 
tor of  which  are  whole  numbers,  however  great  the  numbers  be 
taken.  Lambert's  proof*  showed,  indeed,  that  the  rectification  and 
the  quadrature  of  the  circle  could  not  be  accomplished  in  one  par- 
ticular simple  way,  but  it  still  did  not  exclude  the  possibility  of  the 
problem  being  solvable  in  some  other  more  complicated  way,"  and 
without  requiring  further  aids  than  straight  edge  and  compasses.    \ 


*  Given  in  Legendre's   Geometry,  in  the  Appendix  to  De  Morgan,  op.  cit.,  p. 
495,  and  in  Rudio,  of.  cit.  —  Tr. 


142  THE  SQUARING  OF  THE  CIRCLE. 

Proceeding  slowly  but  surely  it  was  next  sought  to  discover 
the  essential  properties  which  distinguish  problems  solvable  with 
straight  edge  and  compasses  from  problems  the  construction  of 
which  is  elementarily  impossible,  that  is  by  employing  the  postu- 
lates only.  Slight  reflection  showed,  that  a  problem,  to  be  elemen- 
tarily solvable,  must  always  be  such  that  the  unknown  lines  of  its 
figure  are  connected  with  the  known  lines  by  an  equation  for  the 
solution  of  which  equations  of  the  first  and  second  degree  only  are 
requisite,  and  which  can  be  so  arranged  that  the  measures  of  the 
known  lines  will  appear  as  integers  only.  The  conclusion  to  be 
drawn  from  this  was  that  if  the  quadrature  of  the  circle  and  conse- 
quently its  rectification  were  solvable  elementarily,  the  number  n, 
which  represents  the  ratio  of  the  unknown  circumference  to  the 
known  diameter,  must  be  the  root  of  a  certain  equation,  of  a  very 
high  degree  perhaps,  but  in  which  all  the  numbers  are  whole  num- 
bers ;  that  is,  there  would  have  to  exist  an  equation,  made  up  en- 
tirely of  whole  numbers,  which  would  be  correct  if  its  unknown 
quantity  were  made  equal  to  n. 

Since  the  beginning  of  this  centur}%  consequently,  the  efforts 
of  a  number  of  mathematicians  have  been  bent  upon  proving  that 
71  generally  is  not  algebraical,  that  is,  that  it  cannot  be  the  root  of 
an  equation  having  whole  numbers  for  coefficients.  But  mathe- 
matics had  to  make  tremendous  strides  forward  before  the  means 
■were  at  hand  to  accomplish  this  demonstration.  After  the  French 
Academician,  Professor  Hermite,  had  furnished  important  prepara- 
tory assistance  in  his  treatise  Sur  la  Fonction  Expoticntiellc,  pub- 
lished in  the  seventy-seventh  volume  of  the  Comptes  Rendiis,  Pro- 
fessor Lindemann,  at  that  time  of  Freiburg,  now  of  Munich,  finally 
succeeded,  in  June  1882,  in  rigorously  demonstrating  that  the  num- 
ber n  is  not  algebraical,*  and  so  supplied  the  first  proof  that  the 

~fr.  ' 

_.  *For  the  benefit  of  my  mathematical  readers  I  shall  present  here  the  most 
important  steps  of  Lindemann's  demonstration.  M.  Hermite  in  order  to  prove  the 
transcendental  character  of 

,,1,1,1  ,     1 , 

developed  relations  between  certain  definite  integrals  {Comftes  Re^idits  of  the 
Paris  Academy,  Vol.  77,  1873).     Proceeding  from   the  relations   thus  established. 


THE   SQUARING  OF  THE  CIRCLE.  I43 

problems  of  the  rectification  and  squaring  of  the  circle  with  the 
help  only  of  algebraical  instruments  like  straight  edge  and  com- 
passes are  insolvable.  Lindemann's  proof  appeared  successively 
in  the  Reports  of  the  Berlin  Academy  (June,  1882),  in  the  Comptes 
Rendus  of  the  French  Academy  (Vol.  115,  pp.  72  to  74),  and  in  the 
Mathematische  Annalen  (Vol.  20,  pp.  213  to  225). 

"It  is  impossible  with  straight  edge  and  compasses  to  con-  \ 
struct  a  square  equal  in  area  to  a  given  circle."  These  are  the  ' 
words  of  the  final  determination  of  a  controversy  which  is  as  old  as 
the  history  of  the  human  mind.  But  the  race  of  circle-squarers, 
unmindful  of  the  verdict  of  mathematics,  the  most  infallible  of 
arbiters,  will  never  die  out  as  long  as  ignorance  and  the  thirst  for 
glory  remain  united. 


Professor  Lindemann  first  demonstrates  the  following  proposition  :  If  the  coefiB- 
cients  of  an  equation  of  the  «th  degree  are  all  real  or  complex  whole  numbers  and 
the  n  roots  of  this  equation  z^,  z^,  .  .  .,  z„  are  different  from  zero  and  from  each 
other  it  is  impossible  for 

to  be  equal  to  ^,  where  a  and  b  are  real  or  complex  whole  numbers.  It  is  then 
shown  that  also  between  the  functions 

where  r  denotes  an  integer,  no  linear  equation  can  exist  with  rational  ccefficients 
different  from  zero.  Finally  the  beautiful  theorem  results  :  If  2^  is  the  root  of  an 
irreducible  algebraic  equation  the  coefficients  of  which  are  real  or  complex  whole 
numbers,  then  e^  cannot  be  equal  to  a  rational  number.  Now  in  reality  e"^~^  is 
equal  to  a  rational  number,  namely,  — i.  Consequently,  ttV — i,  and  therefore  ~ 
itself,  cannot  be  the  root  of  an  equation  of  the  wth  degree  having  whole  numbers 
for  coefficients,  and  therefore  also  not  of  such  an  equation  having  rational  coeffi- 
cients. The  property  last  mentioned,  however,  rr  would  have  if  the  squaring  of  the 
circle  with  straight  edge  and  compasses  were  possible.  [The  questions  involved 
in  the  discussions  of  the  last  three  pages  have  been  excellently  treated  by  Klein  in 
Fa7nous  Problems  of  Elementary  Geometry  recently  translated  by  Beman  and 
Smith  (Ginn  &  Co.,  Boston).  Lindemann's  proof  is  here  presented  in  a  simplified 
form,  and  so  brought  within  the  comprehension  of  students  conversant  only  with 
algebra. — Tr.\ 


INDEX. 


Abstraction,  95. 

Academy,  French,  112. 

Additicii,  76 ;    associative   law   of,    10 ; 

commutative  law  of,  9. 
Ahmes,  122. 

Alchwarizmi,Muhammed  Ibn  Musa,i3i. 
Algebraic  curves,  theory  of,  35. 
Alhaitam,  Ibn,  131. 
Analytical  geometry,  67. 
Anaxagoras,  123. 
Antiphon,  125. 
Apollonius,  27,  34. 

Approximation,  constructions  of,  137. 
Arabs,  magic  squares  of,   42  ;  squaring 

of  the  circle  among,  128  et  seq. 
Archimedes,  27,  113,  117,  126  et  seq. 
Arithmetic,  36;  pure,  8  et  seq.;  monism 

in,   8-26. 
Aryabhatta,  129. 
Association,  laws  of,  18. 
Associative  law  of  addition,  10. 
Augend,  10. 

Axioms,  Euclid's,  79  et  seq. 
Aztecs,  5,  73. 

Babylonians,    squaring   of   the   circle 

among,  122  et  seq. 
Bacteriology,  36. 
Basselin,  136.- 
Beman,  W.  W.,  143. 
Bhaskara,  129. 
Bible,  speaks  of  four  dimensions,    106  ; 

squaring  of  the  circle  in  the,  123. 
Bischoff,  Professor,  136. 
Bohm,  137. 
Bolyai,  37,  79. 


Bovillius,  132. 
Brahmagupta,  130. 
Brouncker,  Lord,  138. 
Bryson,  125. 

Cagliostro,  102. 

Calculus,  Differential,  29. 

Cardan,  13. 

Cards,  problem  at,  51  et  seq. 

Chemistry,  37;  application  of  the  idea  of 
multi-dimensioned  space  to  theoret- 
ical, 88. 

Chess-knight,  magic  squares  that  in- 
volve the  move  of  the,  57  et  seq. 

Chinese,  130;  squaring  of  the  circle 
among,  128  et  seq. 

Christian  nations,  squaring  of  the  circle 
among,  128  et  seq. 

Cipher,  13. 

Circle-squarers,  113  et  seq. 

Circumversion,  congruence  by,  81. 

Clairvoyance,  103. 

Clavius,  133. 

Clebsch,  Alfred,  35. 

Clerget,  136. 

Clocks,  4. 

Commutation,  15-18. 

Commutative  law  of  addition,  9. 

Complex  numbers,  24,  35. 

Concentric  magic  squares,  55  et  seq. 

Configurations,  88. 

Congruent  figures,  80. 

Conic  sections,  34. 

Construction,  problems  of,  118. 

Continuous  functions,  28. 

Copernicus,  94. 


146 


Corsonich,  Lieutenant  Colonel,  136. 
Counting,  i  et  seq.,  9,  72  et  seq. 
Cranz,  65,  loi. 
Culvasutras,  129. 
Cumberland,  iii. 

Curvature,    negative,    81   et   seq.;    posi- 
tive, 81  et  seq.;  zero,  82  ;  of  space,  85. 
Cyclometry,  127  et  seq. 

Dase,  Zacharias,  139. 

Davis,  the  American  visionary,  102. 

De  Cusa,  Nicolas,  131. 

Definitional  formulae,  11,  21. 

Degrees,  128. 

Degree,  numbers  of,  19. 

De  la  Hire,  45,  46,  51,  53. 

De  Meziriac,  Bachet,  45,  46. 

De  Morgan,  41,  113,  140,  141. 

Depth,  94. 

Descartes,  67. 

De  Serres,  Oliver,  136. 

Detraction,  11. 

Difference,  11. 

Differential  forms,  13  et  seq.,  74. 

Dimension,  concept  of,  65  et  seq. 

Dinostratus,  124. 

Displacement,  congruence  by,  81. 

Distance  of  objects,  96. 

Dumas,  Alexander,    102. 

Duplication  of  the  cube,  30. 

Du  Prel,  Carl,  105. 

Diirer,  Albert,  40,  43,  45. 

Egyptians,    the   squaring   of  the   circle 

among,  122  et  seq. 
Eisenlohr,  29,  122. 
Equations  between  x,  y,  and  z,  70. 
Ether,  75  et  seq. 
Etruscans,  3. 
Euclid,  28,  37,  78,  126. 
Euclid's  axioms,  79-82. 
Euler,  117,  137. 
Evolution,  20,  21. 
Extension  of  notions  in  mathematics,  75 

et  seq. 

Finaeus,  Orontius,  132. 

Fingers,  i   et  seq.  ;  in  reckoning,   4   et 

seq. 
Five  as  a  numeral  basis,  5. 


Foundation-principle,  14. 
Four-dimensional    point-aggregates,    72 

et  seq. 
Four-dimensioned   space,  refutation   of 

the  existence  of,  89  et  seq. 
Fourth  degree,  operation  of,  76. 
Fourth  dimension,  the,  64-1 11. 
Four  variable  quantities,  77. 
Fractional  numbers,  negative,  18. 
Fractions,  17  et  seq. 
Functions,  theory  of,  35. 

Ganefa,  129. 

Gauss,  37,  79,  99. 

Geometry,  37;  origin  of,  33. 

Glaukon,  98. 

Grassmann,  g. 

Greeks,    3,    13  ;  squaring   of   the  circle 

among,  115,  122  et  seq. 
Gregory  of  St.  Vincent,  133. 
Gregory,  James,  135,  138. 
Gudian  manuscript,  128. 

Halley,  139. 

Hamlet,  41. 

Hankel,  9,  14. 

Harmony  of  the  spheres,  63. 

Have,  Mr.,  loi. 

Helmholtz.  37,  79. 

Hermite,  Professor,  142. 

Herodotus,  33. 

Hesse,  136. 

Hindus,  13;  mathematical  performances 

of  the,    128  et  seq.;  squaring  of  the 

circle  among,  128  et  seq. 
Hippias,  123,  124. 
Hippocrates,  125. 
History,  33. 
Hobbes,  136. 
Homer,  84. 
Huge!,  61. 
Hume,  39. 
Huygens,  135,  136. 

Imaginary  numbers,  23,  35. 
Impossibility  of  demonstration,  116. 
Increment,  10. 
Infinitely  great,  ig,  85. 
Insoluble  problems,  30,  143. 
Involution,  19,  21,  76. 
Irrational  numbers,  23. 


INDEX. 


M7 


Jones,  W.,  117. 

Kant,  39,  98  et  seq. 

Kepler,  34. 

Kircher,  Athanasius,  45. 

Klein,  Felix,  31,  143. 

Knight,  move  of  a,  52. 

Knight-problem,  58. 

Knots,  experiment  with  the,  103,  105. 

Kochansky,  61,  137. 

Kronecker,  7,  g. 

Lagny,  139. 

Lambert,  32,  141. 

Law  of  association,  15. 

Legendre,  80,  141. 

Leibnitz,  28,  138. 

Leistner,  136. 

Lessing,  30. 

Liger,  136. 

Limits,  125. 

Lindemann,  31,  116,  142. 

Livy,  2. 

Lobachevski,  37,  79. 

Local  value,  Hindu  system  of,  3. 

Logarithm,  finding  of  the,  20. 

Logarithmand,  21. 

Longomontanus,  133. 

Ludolf,  137. 

Lunes  of  Hippocrates,  126. 

Luther,  loi. 

Machin,   139. 

Magic  cubes,  61  et  seq. 

Magic  Squares,  39-63 ;  problem  and 
origin  of,  42  ;  concentric,  55  et  seq ; 
even-numbered,  49  et  seq. ;  odd-num- 
bered, 44  et  seq, ;  with  magical  parts, 

56  et  seq.;  whose  summation  gives 
the  number  of  a  year,  53  et  seq. ;  that 
involve  the  move  of  the  chess-knight, 

57  et  seq. 

Mathematical  knowledge,  on  the  nature 
of,  27-38  ;  characteristics  of,  35 ;  in- 
trinsic character  of,  27. 

Mathematics,  39 ;  most  conservative  of 
all  sciences,  27;  self-sufficiency  of,  32. 

Mathulon,   136. 

Measuring,  17. 


Mechanical    instruments   of    geometry, 

120  et  seq. 
Mechanics,  38. 
Mediums,  107. 
Melancholy,  40,  41. 
Merkel,  136. 

Methods,  discovery  of,  35. 
Metius,  Peter,  132,  133. 
Mexico,  ancient,  3. 
Mill,  J.  S.,  39. 
Milton,  41. 
Minerva,  3. 
Minus  sign,  11. 
Minutes,  128. 

Models  of  three-dimensional  nets,  78. 
Monism  in  arithmetic,  8-26. 
Monist,  The,  39. 
Moschopulus,   45. 
Multi-dimensional   magnitudes   and 

spaces,  67  et  seq.,  77. 
Multiplication,  15,  77. 
Mysticism,  39. 

Named  number,  6. 

N-dimensional  totalities,  69  et  seq. 

Negative  exponents,  20,  21. 

Negative  numbers,  12  et  seq.,  16,  18.  74. 

Newto'n,  28,  34,  138. 

Nonius,  Petrus,  132. 

Notation,  systems  of,  6. 

Number,  notation  and  definition  of,  1-7; 

denominate,    4 ;    abstract,  7 ;    as    the 

result  of  counting,  9  ;  complex,  24,  35; 

forms  of,  14. 
Numbers,  as  symbols,  7;  imaginary,  23, 

35;  irrational,  23;  real,  23. 
Numbering,  6. 
Number-pictures  and  signs,   natural,   2 

et  seq. 
Number-words,  natural,  4. 
Numerals,  the  formation  of,  5  et  seq. 
Numeral  words,  4  et  seq.;  compound,  4; 

in  the  Indo-Germanic  tongues,  5. 
Numeral  writing,  2  et  seq. 
Numeration,  i  et  seq. 
Numeri  ficti,  13. 
Numeri  veri,  13. 

Pappus,  124. 

Parallels,  theory  of,  28,  79  et  seq. 


148 


INDEX. 


Parting,  17. 

Perception,  95. 

Physics,  8,  33,  38. 

TT,  iiyetseq.,  127,  138  et  seq. 

Plato,  39,  98. 

Polygons,  magical,  59  et  seq. 

Porta,  John,  133. 

Postulates,  121. 

Powers,  20. 

Prime  numbers,  problem  of  the  fre- 
quency of,  31. 

Principle  of  no  exception,  14. 

Prizes  offered  circle-squarers,  116. 

Probabilities,  141. 

Problems,  fundamental,  120;  solvable 
with  straight  edge  and  compasses,  142. 

Ptolemy,  32,  128,  129. 

Pure  arithmetic,  8  et  seq. 

Pythagoras,  27,  123,  136. 

Quadratrix,  124. 

Quadrature  of  the  circle,  in  Egypt,  114; 

□  stands  and  falls  with  the  problem  of 

rectification,    119;   numerical,    117   et 

seq. 
Quantities,   four  variable,  77;   negative, 

12  et  seq. 
Quaternions,  25. 

Radicand,  20. 

Rays,  69. 

Real  numbers,  23. 

Rectification    of   the  circle,  numerical, 

117  et  seq. 
Regiomontanus,  132. 
Rhind  Papyrus,  29,  122. 
Riemann,  37,  79,  100. 
Riese,  Adam,  45. 
Romans,  3,  73;   squaring   of   the   circle 

among,  128  et  seq. 
Romanus,  Adrianus,  133,  134. 
Rudio,  127,  141. 

Sauveur,  45,  61. 
Scaliger,  Joseph,  133. 
Scheffler,   45,  46,  59,  61. 
Schlegel,  65,  78. 
Schroder,  E.,  9,  11. 
Science,   iii. 


Seconds,  128. 

Series,  infinite,  133  et  seq.  ;  Leibnitz's, 
138. 

Shakespeare,  41. 

Shanks,  139, 

Sharp,  Abraham,  139. 

Shortest  line,  82. 

Slade,  loi. 

Smith,  D    E.,  143. 

Snell,  134. 

Socrates,  98. 

Solvability  of  problems,  121. 

Sophists,  124. 

Space,  curvature  of,  85  ;  four-dimen- 
sional, 64  ;  intuition  of,  94  ;  of  expe- 
rience, 83  ;  of  multiple  dimensions, 
great  service  to  science,  85. 

Spheres  in  space,  70  et  seq. 

Spirits,  existence  of  four-dimensional,  98 
etseq.;  of  the  dead,  65. 

Spirit-rappings,  101. 

Spirit-writing,  loi. 

Spiritualistic,    mediums,    90   et   seq.; 
seance,  in. 

Spiritualists,  64. 

Squaring  of  the  circle,  30,  1 12-143, 

Squares  in  space,  71. 

Stifel,  Michael,  45. 

Straight  edge  and  compasses,  to  con- 
struct with,  119  et  seq. 

Subtertraction,  11. 

Subtraction,  10  et  seq.,  73  et  seq. 

Subtrahent,  11. 

Summands,  9. 

Supernatural  influences,  no  et  seq. 

Sylvester  II.,  131. 

Symbols,  2. 

Symmetry,  91. 

Talmud,  123. 

Ten,  4. 

rErpayo)Viil^s/.v,  115. 

Thales,  123. 

Thing-in-itself,  98. 

Third  degree,  operations  of  the,  19. 

Three  Bodies,  problem  of,  31. 

Time,  36. 

Triangle,  sum  of  the  angles  of  a,  83. 

Trisection  of  the  angle,  30. 

Twenty,  as  base,  5. 


149 


TJlrici,  107. 
Ulysses,  84. 
Unlimited,  85. 
Unnamed  number,  6. 


Van  Ceulen,  Ludolf,  134. 
Van  Eyck,  Simon,  132. 
Van't  Hoff,  88. 
Vicenary  system,  5, 
Vieta,   133. 
Vision,  93  et  seq.  ' 
Vitruvius,  128. 
Von  Liittich,  Frankos,  131. 


Wallis,  136,  138. 
Waltershausen,  gg. 
Weierstrass,  31. 
Wenzelides,  58. 
Wislicenus,   88. 
Wolfenbuttel  Library,  128. 
Wolff,  Professor,   140. 
Words,  numeral,  4  et  seq. 
Worms,  two-dimensional,  104. 
Wundt,  107. 

Zero,  13,  16. 

Zero  exponents,  21. 

Zollner,  65,  92,  g3  et  seq. 


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